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This document is an introduction to combinatorics presented by A.B. Benedict Balbuena from the University of the Philippines. It discusses fundamental combinatorics concepts like the addition rule, product rule, and inclusion-exclusion principle. Examples of counting problems are provided to illustrate how to use these rules to calculate the number of possible outcomes in situations involving sets, permutations, and combinations.

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Combinatorics

This document provides an introduction to combinatorics including the sum rule, product rule, permutations, and combinations. The sum rule states that if tasks are independent, the number of ways to do either task is the sum of the number of ways to do each individually. The product rule states that if tasks are independent, the number of ways to do both tasks is the product of the number of ways to do each. Permutations refer to ordered arrangements and combinations refer to unordered arrangements. The document includes examples of applying these concepts.

Combinatorics

Combinatorics is a subfield of discrete mathematics that focuses on counting combinations and arrangements of discrete objects. It involves counting the number of ways to put things together into various combinations. Some key rules in combinatorics include the sum rule, which states that the number of ways to accomplish either of two independent tasks is the sum of the number of ways to accomplish each task individually. The product rule states that the number of ways to accomplish two independent tasks is the product of the number of ways to accomplish each task. Generating functions can be used to efficiently represent counting sequences by coding terms as coefficients of a variable in a formal power series. They allow problems involving counting and arrangements to be solved using operations with formal power series.

Principle of mathematical induction

The document discusses the principle of mathematical induction and how it can be used to prove statements about natural numbers. It provides examples of using induction to prove statements about sums, products, and divisibility. The principle of induction states that to prove a statement P(n) is true for all natural numbers n, one must show that P(1) is true and that if P(k) is true, then P(k+1) is also true. The document provides examples of direct proofs of P(1) and inductive proofs of P(k+1) to demonstrate applications of the principle.

Hasse diagram

The document defines and provides examples of Hasse diagrams. A Hasse diagram is a type of graph used to represent partially ordered sets. It draws elements with edges between them if one element covers another. The document gives an example Hasse diagram and explains that it shows the relations between elements in a partially ordered set with edges between elements if one is directly above the other in the order.

Set in discrete mathematics

Discrete Mathematics - Sets. ... He had defined a set as a collection of definite and distinguishable objects selected by the means of certain rules or description. Set theory forms the basis of several other fields of study like counting theory, relations, graph theory and finite state machines.

Logic (PROPOSITIONS)

This document discusses propositional logic and truth tables. It defines primitive and compound propositions. Logical connectives like negation, disjunction, and conjunction are explained. Propositional variables are used to represent statements that can be true or false. Truth tables list all possible combinations of true and false values for propositional variables and determine the truth value of compound statements formed from logical connectives. The number of rows in a truth table is determined by 2 to the power of the number of propositional variables. Several examples of truth tables are given for logical connectives like negation, disjunction, conjunction, implication, and biconditional.

Fundamental principle of counting- ch 6 - Discrete Mathematics

The document discusses the generalized product rule for determining the number of ways a procedure consisting of sequential tasks can be carried out. It states that if there are n1 ways to complete task 1, n2 ways to complete task 2, and so on up to nm ways to complete the final task, then the total number of ways to complete the entire procedure is the product of n1, n2, ..., nm. It provides the example that in a class of 60 students, there are at least 12 students who will receive the same letter grade of A, B, C, D, or F.

Combinatorics.pptx

This document provides an introduction to the topic of combinatorics. It discusses enumeration and graph theory as key aspects of combinatorics. Examples are given of factorial expressions, combinations, permutations, and sequences like arithmetic and geometric progressions. Common applications mentioned include loops and edges in graphs, as well as real world networks like maps, the internet, and social media.

Combinatorics

This document provides an introduction to combinatorics including the sum rule, product rule, permutations, and combinations. The sum rule states that if tasks are independent, the number of ways to do either task is the sum of the number of ways to do each individually. The product rule states that if tasks are independent, the number of ways to do both tasks is the product of the number of ways to do each. Permutations refer to ordered arrangements and combinations refer to unordered arrangements. The document includes examples of applying these concepts.

Combinatorics

Combinatorics is a subfield of discrete mathematics that focuses on counting combinations and arrangements of discrete objects. It involves counting the number of ways to put things together into various combinations. Some key rules in combinatorics include the sum rule, which states that the number of ways to accomplish either of two independent tasks is the sum of the number of ways to accomplish each task individually. The product rule states that the number of ways to accomplish two independent tasks is the product of the number of ways to accomplish each task. Generating functions can be used to efficiently represent counting sequences by coding terms as coefficients of a variable in a formal power series. They allow problems involving counting and arrangements to be solved using operations with formal power series.

Principle of mathematical induction

The document discusses the principle of mathematical induction and how it can be used to prove statements about natural numbers. It provides examples of using induction to prove statements about sums, products, and divisibility. The principle of induction states that to prove a statement P(n) is true for all natural numbers n, one must show that P(1) is true and that if P(k) is true, then P(k+1) is also true. The document provides examples of direct proofs of P(1) and inductive proofs of P(k+1) to demonstrate applications of the principle.

Hasse diagram

The document defines and provides examples of Hasse diagrams. A Hasse diagram is a type of graph used to represent partially ordered sets. It draws elements with edges between them if one element covers another. The document gives an example Hasse diagram and explains that it shows the relations between elements in a partially ordered set with edges between elements if one is directly above the other in the order.

Set in discrete mathematics

Discrete Mathematics - Sets. ... He had defined a set as a collection of definite and distinguishable objects selected by the means of certain rules or description. Set theory forms the basis of several other fields of study like counting theory, relations, graph theory and finite state machines.

Logic (PROPOSITIONS)

This document discusses propositional logic and truth tables. It defines primitive and compound propositions. Logical connectives like negation, disjunction, and conjunction are explained. Propositional variables are used to represent statements that can be true or false. Truth tables list all possible combinations of true and false values for propositional variables and determine the truth value of compound statements formed from logical connectives. The number of rows in a truth table is determined by 2 to the power of the number of propositional variables. Several examples of truth tables are given for logical connectives like negation, disjunction, conjunction, implication, and biconditional.

Fundamental principle of counting- ch 6 - Discrete Mathematics

The document discusses the generalized product rule for determining the number of ways a procedure consisting of sequential tasks can be carried out. It states that if there are n1 ways to complete task 1, n2 ways to complete task 2, and so on up to nm ways to complete the final task, then the total number of ways to complete the entire procedure is the product of n1, n2, ..., nm. It provides the example that in a class of 60 students, there are at least 12 students who will receive the same letter grade of A, B, C, D, or F.

Combinatorics.pptx

This document provides an introduction to the topic of combinatorics. It discusses enumeration and graph theory as key aspects of combinatorics. Examples are given of factorial expressions, combinations, permutations, and sequences like arithmetic and geometric progressions. Common applications mentioned include loops and edges in graphs, as well as real world networks like maps, the internet, and social media.

Discrete mathematics Ch2 Propositional Logic_Dr.khaled.Bakro د. خالد بكرو

Discrete Mathematics chapter 2 covers propositional logic. A proposition is a statement that is either true or false. Propositional logic uses propositional variables and logical operators like negation, conjunction, disjunction, implication and biconditional. Truth tables are used to determine the truth value of compound propositions formed using these operators. Logical equivalences between compound propositions can be shown using truth tables or by applying equivalence rules.

Proofs by contraposition

The document discusses proofs by contraposition. It explains that a statement of the form "if p then q" can be proven by showing its contrapositive "if not q then not p" is true. It provides examples of proofs using this method, including proving if n^2 is even then n is even, if m + n is even then m and n have the same parity, and if 3n + 2 is odd then n is odd. Homework exercises are provided applying this proof technique.

Discrete Mathematics Lecture

The document discusses key concepts in discrete mathematics and logic. It defines propositions as basic building blocks represented by letters, and connectives as operators used to combine propositions like conjunction, disjunction, negation, and conditional statements. It provides truth tables showing the truth values of propositions under different connectives and examples of applying these logic rules.

Predicates and quantifiers

This document introduces predicates and quantifiers in predicate logic. It defines predicates as functions that take objects and return propositions. Predicates allow reasoning about whole classes of entities. Quantifiers like "for all" (universal quantifier ∀) and "there exists" (existential quantifier ∃) are used to make general statements about predicates over a universe of discourse. Examples demonstrate how predicates and quantifiers can represent concepts like "all parking spaces are full" or "some parking space is full." Laws of quantifier equivalence and negation rules with quantifiers are also presented.

Unit I discrete mathematics lecture notes

The Foundations: Logic and Proofs: Propositional Logic, Applications of Propositional Logic, Propositional Equivalence, Predicates and Quantifiers, Nested Quantifiers, Rules of Inference, Introduction to Proofs, Proof Methods and Strategy.

Poset in Relations(Discrete Mathematics)

The document discusses partial ordered sets (POSETs). It begins by defining a POSET as a set A together with a partial order R, which is a relation on A that is reflexive, antisymmetric, and transitive. An example is given of the set of integers under the relation "greater than or equal to". It is shown that this relation satisfies the three properties of a partial order. The document emphasizes that a relation must satisfy all three properties - reflexive, antisymmetric, and transitive - to be considered a partial order. Some example relations on a set are provided and it is discussed which of these are partial orders.

CMSC 56 | Lecture 1: Propositional Logic

Propositional Logic
CMSC 56 | Discrete Mathematical Structure for Computer Science
August 17, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas

Discrete mathematic

The document discusses the key topics in discrete mathematics that will be covered across 5 units. Unit 1 covers sets, relations, functions and their properties. Unit 2 discusses mathematical induction, counting techniques, and number theory topics. Unit 3 is on propositional logic, logical equivalence and proof techniques. Unit 4 covers algebraic structures like groups, rings and fields. Unit 5 is about graphs, trees, and their properties like coloring and shortest paths. The document also lists 3 recommended textbooks for the course.

The Fundamental Counting Principle

Subway and Starbucks advertise over 10,000 combinations of ways to customize sandwiches and drinks due to the many condiment and serving options. Companies can calculate combinations using (1) tree diagrams to show all outcomes or (2) the fundamental counting principle of multiplying the number of options at each choice point. Tree diagrams specifically show individual outcomes while the counting principle gives the total number of outcomes. Examples demonstrate using each method to find combinations in probability problems.

Logical equivalence, laws of logic

Logical Equivalence
Laws of Logic
Proving logical equivalences using Laws of Logic and Truth Tables

Modular arithmetic

This document defines modular arithmetic and some of its key properties. Modular arithmetic involves taking the remainder of dividing one integer by another. It defines congruence (a ≡ b mod n) as meaning n divides the difference of a and b. Some key properties are that addition, subtraction and multiplication of congruent numbers are also congruent modulo n, and that modular arithmetic forms a mathematical system with properties like commutativity, associativity and distributivity. Examples are given to illustrate these concepts and properties.

Discrete Structures. Lecture 1

This document outlines a course on discrete structures that covers topics like logic, proofs, sets, relations, graphs and trees. It begins with an introduction that distinguishes between discrete and continuous data. It then defines discrete mathematics as the study of discrete objects and structures. The syllabus lists the topics to be covered in the course. Reference books are provided and the document proceeds to provide examples and explanations of concepts like propositions, logical connectives, truth tables and how to form compound propositions using logical operators.

Chapter 1 Logic of Compound Statements

This document introduces basic concepts in propositional logic and discrete mathematics including:
- Statements and their truth values
- Logical connectives such as negation, conjunction, disjunction, implication, biconditional
- Compound statements formed using logical connectives
- Truth tables to determine the truth values of compound statements
- Tautologies, contradictions and contingencies
- Negation, contrapositive, converse and inverse of conditional statements
- De Morgan's laws of negation for conjunction and disjunction
Examples are provided to illustrate key concepts and definitions throughout.

Mathematical induction by Animesh Sarkar

The document discusses various mathematical induction concepts including weak induction, strong induction, and examples of proofs using induction. It begins with an overview of weak induction and its two steps: proving a statement is true for the base case, and proving if it is true for some step k then it is true for k+1. Strong induction is then introduced which proves a statement is true for all steps up to k implies it is true for k+1. Examples are provided of proofs using induction, such as proving 3n-1 is divisible by 2 and that sums of odd numbers equal squares. Reference sources are listed at the end.

Number theory

This document introduces some basic concepts in number theory, including primes, least common multiples, greatest common divisors, and modular arithmetic. It then discusses different number systems such as decimal, binary, octal, and hexadecimal. Methods are provided for converting between these different bases, including dividing or grouping bits and multiplying by the place value. Prime numbers are defined as integers greater than 1 that are only divisible by 1 and themselves. The fundamental theorem of arithmetic states that every integer can be written as a unique product of prime factors.

Recursion DM

The document discusses recursion and recursively defined sequences, functions, and algorithms. It provides examples of recursively defined sequences like powers of 2 and the Fibonacci sequence. Recursively defined functions are given, such as factorials and Fibonacci numbers. Recursive algorithms solve problems by reducing them to smaller instances of the same problem, like the Euclidean algorithm to find the greatest common divisor or a recursive Fibonacci algorithm. While recursive solutions are often shorter and more elegant, iterative algorithms using loops are usually more efficient.

Number theory

This document provides information about number theory, including divisors, prime factorization, and congruences. It begins by defining divisors and the division algorithm, and proves several theorems about greatest common divisors and expressing them as linear combinations. It then discusses prime numbers and Euclid's lemma, and proves the fundamental theorem of arithmetic that every integer can be uniquely expressed as a product of prime factors. The document concludes by defining congruences modulo m and listing some basic properties of congruences.

Mathematical induction

Mathematical induction is a method of proof that can be used to prove that a statement is true for all positive integers. It involves two steps: 1) proving the statement is true for the base case, usually n = 1, and 2) assuming the statement is true for an integer k and using this to prove the statement is true for k + 1. Examples are provided to demonstrate how to use mathematical induction to prove statements such as the sum of the first n positive integers equalling n(n+1)/2 and that 7n - 1 is divisible by 6 for all positive integers n.

Pigeonhole Principle,Cardinality,Countability

The document discusses the pigeonhole principle, countability, and cardinality. The pigeonhole principle states that if n items are placed into m pigeonholes with n > m, then at least one pigeonhole must contain more than one item. Countability refers to sets being either finite or denumerable (having the same cardinality as the natural numbers). Cardinality compares the sizes of sets based on whether a bijection exists between them. The document provides examples and proofs of these concepts.

CMSC 56 | Lecture 16: Equivalence of Relations & Partial Ordering

Equivalence of Relations & Partial Ordering
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 21, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas

Partitions of a number sb

A partition of a positive integer is a way of writing that integer as a sum of positive integers. The document provides an example of the 5 partitions of the number 4: 4, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1. It asks the reader to find all 15 partitions of the number 7, listing them out.

2012 scte presentation_lsc_updated_2

The document discusses effective ways to retain members through each step of the membership lifecycle. It outlines the 5 steps as awareness, recruitment, engagement, renewal, and reinstatement. It emphasizes that retention begins before recruitment by building awareness and promoting the organization's value. During engagement and renewal, members must feel they are receiving value to remain confident in the organization. Communication, benefits, and a sense of community encourage members to renew their membership.

Discrete mathematics Ch2 Propositional Logic_Dr.khaled.Bakro د. خالد بكرو

Discrete Mathematics chapter 2 covers propositional logic. A proposition is a statement that is either true or false. Propositional logic uses propositional variables and logical operators like negation, conjunction, disjunction, implication and biconditional. Truth tables are used to determine the truth value of compound propositions formed using these operators. Logical equivalences between compound propositions can be shown using truth tables or by applying equivalence rules.

Proofs by contraposition

The document discusses proofs by contraposition. It explains that a statement of the form "if p then q" can be proven by showing its contrapositive "if not q then not p" is true. It provides examples of proofs using this method, including proving if n^2 is even then n is even, if m + n is even then m and n have the same parity, and if 3n + 2 is odd then n is odd. Homework exercises are provided applying this proof technique.

Discrete Mathematics Lecture

The document discusses key concepts in discrete mathematics and logic. It defines propositions as basic building blocks represented by letters, and connectives as operators used to combine propositions like conjunction, disjunction, negation, and conditional statements. It provides truth tables showing the truth values of propositions under different connectives and examples of applying these logic rules.

Predicates and quantifiers

This document introduces predicates and quantifiers in predicate logic. It defines predicates as functions that take objects and return propositions. Predicates allow reasoning about whole classes of entities. Quantifiers like "for all" (universal quantifier ∀) and "there exists" (existential quantifier ∃) are used to make general statements about predicates over a universe of discourse. Examples demonstrate how predicates and quantifiers can represent concepts like "all parking spaces are full" or "some parking space is full." Laws of quantifier equivalence and negation rules with quantifiers are also presented.

Unit I discrete mathematics lecture notes

The Foundations: Logic and Proofs: Propositional Logic, Applications of Propositional Logic, Propositional Equivalence, Predicates and Quantifiers, Nested Quantifiers, Rules of Inference, Introduction to Proofs, Proof Methods and Strategy.

Poset in Relations(Discrete Mathematics)

The document discusses partial ordered sets (POSETs). It begins by defining a POSET as a set A together with a partial order R, which is a relation on A that is reflexive, antisymmetric, and transitive. An example is given of the set of integers under the relation "greater than or equal to". It is shown that this relation satisfies the three properties of a partial order. The document emphasizes that a relation must satisfy all three properties - reflexive, antisymmetric, and transitive - to be considered a partial order. Some example relations on a set are provided and it is discussed which of these are partial orders.

CMSC 56 | Lecture 1: Propositional Logic

Propositional Logic
CMSC 56 | Discrete Mathematical Structure for Computer Science
August 17, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas

Discrete mathematic

The document discusses the key topics in discrete mathematics that will be covered across 5 units. Unit 1 covers sets, relations, functions and their properties. Unit 2 discusses mathematical induction, counting techniques, and number theory topics. Unit 3 is on propositional logic, logical equivalence and proof techniques. Unit 4 covers algebraic structures like groups, rings and fields. Unit 5 is about graphs, trees, and their properties like coloring and shortest paths. The document also lists 3 recommended textbooks for the course.

The Fundamental Counting Principle

Subway and Starbucks advertise over 10,000 combinations of ways to customize sandwiches and drinks due to the many condiment and serving options. Companies can calculate combinations using (1) tree diagrams to show all outcomes or (2) the fundamental counting principle of multiplying the number of options at each choice point. Tree diagrams specifically show individual outcomes while the counting principle gives the total number of outcomes. Examples demonstrate using each method to find combinations in probability problems.

Logical equivalence, laws of logic

Logical Equivalence
Laws of Logic
Proving logical equivalences using Laws of Logic and Truth Tables

Modular arithmetic

This document defines modular arithmetic and some of its key properties. Modular arithmetic involves taking the remainder of dividing one integer by another. It defines congruence (a ≡ b mod n) as meaning n divides the difference of a and b. Some key properties are that addition, subtraction and multiplication of congruent numbers are also congruent modulo n, and that modular arithmetic forms a mathematical system with properties like commutativity, associativity and distributivity. Examples are given to illustrate these concepts and properties.

Discrete Structures. Lecture 1

This document outlines a course on discrete structures that covers topics like logic, proofs, sets, relations, graphs and trees. It begins with an introduction that distinguishes between discrete and continuous data. It then defines discrete mathematics as the study of discrete objects and structures. The syllabus lists the topics to be covered in the course. Reference books are provided and the document proceeds to provide examples and explanations of concepts like propositions, logical connectives, truth tables and how to form compound propositions using logical operators.

Chapter 1 Logic of Compound Statements

This document introduces basic concepts in propositional logic and discrete mathematics including:
- Statements and their truth values
- Logical connectives such as negation, conjunction, disjunction, implication, biconditional
- Compound statements formed using logical connectives
- Truth tables to determine the truth values of compound statements
- Tautologies, contradictions and contingencies
- Negation, contrapositive, converse and inverse of conditional statements
- De Morgan's laws of negation for conjunction and disjunction
Examples are provided to illustrate key concepts and definitions throughout.

Mathematical induction by Animesh Sarkar

The document discusses various mathematical induction concepts including weak induction, strong induction, and examples of proofs using induction. It begins with an overview of weak induction and its two steps: proving a statement is true for the base case, and proving if it is true for some step k then it is true for k+1. Strong induction is then introduced which proves a statement is true for all steps up to k implies it is true for k+1. Examples are provided of proofs using induction, such as proving 3n-1 is divisible by 2 and that sums of odd numbers equal squares. Reference sources are listed at the end.

Number theory

This document introduces some basic concepts in number theory, including primes, least common multiples, greatest common divisors, and modular arithmetic. It then discusses different number systems such as decimal, binary, octal, and hexadecimal. Methods are provided for converting between these different bases, including dividing or grouping bits and multiplying by the place value. Prime numbers are defined as integers greater than 1 that are only divisible by 1 and themselves. The fundamental theorem of arithmetic states that every integer can be written as a unique product of prime factors.

Recursion DM

The document discusses recursion and recursively defined sequences, functions, and algorithms. It provides examples of recursively defined sequences like powers of 2 and the Fibonacci sequence. Recursively defined functions are given, such as factorials and Fibonacci numbers. Recursive algorithms solve problems by reducing them to smaller instances of the same problem, like the Euclidean algorithm to find the greatest common divisor or a recursive Fibonacci algorithm. While recursive solutions are often shorter and more elegant, iterative algorithms using loops are usually more efficient.

Number theory

This document provides information about number theory, including divisors, prime factorization, and congruences. It begins by defining divisors and the division algorithm, and proves several theorems about greatest common divisors and expressing them as linear combinations. It then discusses prime numbers and Euclid's lemma, and proves the fundamental theorem of arithmetic that every integer can be uniquely expressed as a product of prime factors. The document concludes by defining congruences modulo m and listing some basic properties of congruences.

Mathematical induction

Mathematical induction is a method of proof that can be used to prove that a statement is true for all positive integers. It involves two steps: 1) proving the statement is true for the base case, usually n = 1, and 2) assuming the statement is true for an integer k and using this to prove the statement is true for k + 1. Examples are provided to demonstrate how to use mathematical induction to prove statements such as the sum of the first n positive integers equalling n(n+1)/2 and that 7n - 1 is divisible by 6 for all positive integers n.

Pigeonhole Principle,Cardinality,Countability

The document discusses the pigeonhole principle, countability, and cardinality. The pigeonhole principle states that if n items are placed into m pigeonholes with n > m, then at least one pigeonhole must contain more than one item. Countability refers to sets being either finite or denumerable (having the same cardinality as the natural numbers). Cardinality compares the sizes of sets based on whether a bijection exists between them. The document provides examples and proofs of these concepts.

CMSC 56 | Lecture 16: Equivalence of Relations & Partial Ordering

Equivalence of Relations & Partial Ordering
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 21, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas

Discrete mathematics Ch2 Propositional Logic_Dr.khaled.Bakro د. خالد بكرو

Discrete mathematics Ch2 Propositional Logic_Dr.khaled.Bakro د. خالد بكرو

Proofs by contraposition

Proofs by contraposition

Discrete Mathematics Lecture

Discrete Mathematics Lecture

Predicates and quantifiers

Predicates and quantifiers

Unit I discrete mathematics lecture notes

Unit I discrete mathematics lecture notes

Poset in Relations(Discrete Mathematics)

Poset in Relations(Discrete Mathematics)

CMSC 56 | Lecture 1: Propositional Logic

CMSC 56 | Lecture 1: Propositional Logic

Discrete mathematic

Discrete mathematic

The Fundamental Counting Principle

The Fundamental Counting Principle

Logical equivalence, laws of logic

Logical equivalence, laws of logic

Modular arithmetic

Modular arithmetic

Discrete Structures. Lecture 1

Discrete Structures. Lecture 1

Chapter 1 Logic of Compound Statements

Chapter 1 Logic of Compound Statements

Mathematical induction by Animesh Sarkar

Mathematical induction by Animesh Sarkar

Number theory

Number theory

Recursion DM

Recursion DM

Number theory

Number theory

Mathematical induction

Mathematical induction

Pigeonhole Principle,Cardinality,Countability

Pigeonhole Principle,Cardinality,Countability

CMSC 56 | Lecture 16: Equivalence of Relations & Partial Ordering

CMSC 56 | Lecture 16: Equivalence of Relations & Partial Ordering

Partitions of a number sb

A partition of a positive integer is a way of writing that integer as a sum of positive integers. The document provides an example of the 5 partitions of the number 4: 4, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1. It asks the reader to find all 15 partitions of the number 7, listing them out.

2012 scte presentation_lsc_updated_2

The document discusses effective ways to retain members through each step of the membership lifecycle. It outlines the 5 steps as awareness, recruitment, engagement, renewal, and reinstatement. It emphasizes that retention begins before recruitment by building awareness and promoting the organization's value. During engagement and renewal, members must feel they are receiving value to remain confident in the organization. Communication, benefits, and a sense of community encourage members to renew their membership.

Counting i (slides)

The document discusses counting principles like the multiplication principle, permutations, and combinations. It provides theorems and examples to explain how to calculate the number of possible outcomes for different counting problems. The multiplication principle states that if one task can be performed in n1 ways and another in n2 ways, doing them sequentially allows for n1n2 outcomes. Permutations and combinations deal with arranging or selecting elements from a set. Permutations account for order while combinations do not. Formulas are given to calculate the number of permutations and combinations for a given problem.

1532 fourier series

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics

Math 1300: Section 7- 4 Permutations and Combinations

The document discusses permutations and combinations. It provides definitions and examples of factorials, permutations of n objects taken r at a time, and combinations of n objects taken r at a time. It includes theorems on calculating the number of permutations and combinations. Examples are provided to demonstrate calculating permutations and combinations for different values of n and r.

Roots of polynomial equations

1) Polynomial equations have as many roots as the highest power of the variable. The roots can be real or complex, and may be repeated.
2) Quadratic equations can be solved by setting the coefficients equal to functions of the roots, or by factorizing the quadratic expression.
3) Cubic equations have three roots that relate to the coefficients, and their symmetrical functions can be written in terms of sums and products of the roots.

4.4 probability of compound events

This document defines and provides examples of simple, compound, mutually exclusive and inclusive events. It explains that for mutually exclusive events, the probability of A or B equals the sum of the individual probabilities, while for inclusive events it equals the sum of the individual probabilities minus the probability they both occur. It then gives examples of calculating probabilities for various card draws and die rolls.

12.4 probability of compound events

The document defines key probability terms like union, intersection, mutually exclusive events, and complement. It provides examples of calculating probabilities of compound events, such as the probability of event A or B. It explains that if events are mutually exclusive, the probability of their union is the sum of their individual probabilities. The document ends with practice problems calculating probabilities of compound events using these concepts.

Arithmetic sequence

This document discusses arithmetic sequences and provides formulas for writing recursive and explicit expressions for terms in an arithmetic sequence. It gives an example of the arithmetic sequence 20, 24, 28, 32, 36, which has a common difference of 4. The recursive formula is defined as an = an-1 + d. The explicit or closed form formula is defined as an = a1 + (n-1)d, which allows finding any term without knowing all previous terms. It asks the reader to find the term number for the last term in the sequence -6, -3, 0, 3, ..., 147 and to determine the values of d and c in the explicit formula an = dn + c for the sequence 8, 6,

Polynomial equations

Here are the remainders when dividing the given polynomials by the specified polynomials:
1. The remainder is 0. Therefore, x-1 is a factor of x3+3x2-4x+2.
2. The remainder is 5.
3. The remainder is 0. Therefore, x+2 is a factor of 2x3+5x2+3x+11.
4. The remainder is 4.
5. The remainder is 7.
6. The remainder is 2.

Arithmetic Sequences

An arithmetic sequence is a sequence where the difference between consecutive terms is constant. The common difference (d) is this constant value. The recursive formula for an arithmetic sequence is Un = Un-1 + d and the explicit formula is Un = U1 + (n - 1)d. Given values for the first few terms, you can find the common difference and write the recursive and explicit formulas to generate all terms of the sequence.

Measures of Position

This document discusses measures of position such as percentile rank and quartiles. It provides examples of calculating percentile rank for a data value and finding the data value that corresponds to a given percentile. Quartiles (Q1, Q2, Q3) divide a data set into quarters. The second quartile is the median. Examples demonstrate finding quartiles for a data set and drawing and interpreting a box plot.

Recursion

Introduction to recursion, recursive function, it's advantages and disadvantages, examples and Tower Of Hanoi (TOH)

Recursion

Recursion is a process where an object is defined in terms of smaller versions of itself. It involves a base case, which is the simplest definition that cannot be further reduced, and a recursive case that defines the object for larger inputs in terms of smaller ones until the base case is reached. Examples where recursion is commonly used include defining mathematical functions, number sequences, data structures, and language grammars. While recursion can elegantly solve problems, iterative algorithms are generally more efficient.

Recursion

This document provides an overview of syntax and generative grammar. It defines syntax as the way words are arranged to show relationships of meaning within and between sentences. Grammar is defined as the art of writing, but is now used to study language. Generative grammar uses formal rules to generate an infinite set of grammatical sentences. It distinguishes between deep structure and surface structure. Tree diagrams are used to represent syntactic structures with symbols like S, NP, VP. Phrase structure rules, lexical rules, and movement rules are discussed. Complement phrases and recursion are also explained.

Permutation & Combination

A short presentation to explain the use of permutations and combinations and some examples to illustrate the concepts. This was made as an assignment in which i was to explain the concepts to the class.

Recursion

Recursion is a technique that involves defining a function in terms of itself via self-referential calls. It can be used to loop without an explicit loop statement. A recursive function must have a base case that does not involve further recursion in order to terminate after a finite number of calls. When a recursive function is called, information like function values and local variables are pushed onto a call stack to keep track of each nested call.

Measures of position

The document contains lyrics from 18 different songs. It discusses a variety of themes related to love and relationships, including being together, being apart, waiting for someone, and remembering past loves and relationships. Locations like "here," "there," and "somewhere" are referenced in several of the songs. The lyrics are in English and Tagalog.

Theories of Composition

This document discusses compositional theories of art including the rule of thirds, golden rectangle, leading lines, lines of sight, strong diagonals, point of view, framing, simplifying, filling the frame, and active space. It provides examples of how various artists have applied these principles in their own works. The goal is to analyze compositions using these established techniques.

Partition Of India

The Partition of India divided British India into two new independent countries - India and Pakistan. On August 14-15, 1947, as Britain withdrew from India, the subcontinent was divided along religious lines into a Hindu-majority India and Muslim-majority Pakistan and Bangladesh. The partition displaced up to 12.5 million people and caused widespread violence, as riots broke out between Hindus and Muslims across the region. Over a million people died in the ensuing violence and chaos of partition.

Partitions of a number sb

Partitions of a number sb

2012 scte presentation_lsc_updated_2

2012 scte presentation_lsc_updated_2

Counting i (slides)

Counting i (slides)

1532 fourier series

1532 fourier series

Math 1300: Section 7- 4 Permutations and Combinations

Math 1300: Section 7- 4 Permutations and Combinations

Roots of polynomial equations

Roots of polynomial equations

4.4 probability of compound events

4.4 probability of compound events

12.4 probability of compound events

12.4 probability of compound events

Arithmetic sequence

Arithmetic sequence

Polynomial equations

Polynomial equations

Arithmetic Sequences

Arithmetic Sequences

Measures of Position

Measures of Position

Recursion

Recursion

Recursion

Recursion

Recursion

Recursion

Permutation & Combination

Permutation & Combination

Recursion

Recursion

Measures of position

Measures of position

Theories of Composition

Theories of Composition

Partition Of India

Partition Of India

The principle of inclusion and exclusion for three sets by sharvari

The document explains the principle of inclusion and exclusion for calculating the cardinality (size) of the union of three sets. It defines the formula that the cardinality of the union of sets A, B and C equals the sum of the cardinalities of the individual sets minus the overlaps between pairs of sets plus the overlap of all three sets. It provides three examples applying this formula to solve for the size of unions of various sets.

2.2 Set Operations

This document discusses set operations and identities. It defines operations like union, intersection, complement, difference, and cardinality. It presents examples of calculating unions, intersections, complements, and differences of sets. It also covers set identities like commutative, associative, distributive, De Morgan's laws, and absorption laws. Methods for proving identities like subset proofs and membership tables are described. An example proof of the second De Morgan's law is provided using subset notation.

Pdm presentation

The document discusses algebraic sets and their properties. It defines algebraic sets as having analogous algebraic properties to arithmetic, with set operations replacing arithmetic operations. It covers the fundamental laws of set algebra, the principle of duality, inclusion-exclusion principle, and using algebra to prove set identities. Examples are provided to illustrate calculating unions and intersections of sets and counting elements that satisfy multiple conditions.

mathematical sets.pdf

The document provides information about sets including:
1) Sets can be finite, infinite, empty, or singleton and are represented using curly brackets. Common set operations are union, intersection, and complement.
2) A set's cardinality refers to the number of elements it contains. Subsets are sets contained within other sets.
3) Examples are given of representing sets using roster and set-builder methods and performing set operations like union, intersection, and complement on sample sets.
4) Basic properties of sets like idempotent, commutative, and associative laws for simplifying set expressions are outlined.

8-Sets-2.ppt

The document discusses set operations such as union, intersection, difference, and complement. It defines each operation formally and provides examples. Properties of each operation are described, such as the commutative, associative, identity and domination laws. Disjoint sets are defined as sets whose intersection is the empty set. The cardinality of the union and intersection of finite sets A and B is discussed. Methods for proving set identities are presented, including using basic set identities, subset proofs, and set builder notation.

20200911-XI-Maths-Sets-2 of 2-Ppt.pdf

The document discusses key concepts about sets, including:
1) Intervals are subsets of real numbers that can be open, closed, or half-open/half-closed. Intervals are represented visually on a number line.
2) The power set of a set A contains all possible subsets of A. Its size is 2 to the power of the size of A.
3) The union of sets A and B contains all elements that are in A, B, or both. The intersection contains all elements common to both sets.
4) Practical problems can use formulas involving the sizes of unions and intersections of finite sets.

2 》set operation.pdf

The document discusses different types of sets and set operations. It defines infinite or uncountable sets as sets whose elements are uncountable and the last member is unknown. It describes overlapping or joint sets as sets that share at least one common element, and disjoint sets as sets with no elements in common. Set operations covered include union, intersection, difference, and complement. Union is the set of all elements that are in set A or B or both. Intersection is the set of common elements in both sets. Difference is the elements in set A that are not in set B. Complement is the elements not included in the original set when taking the difference from the universal set.

CPSC 125 Ch 3 Sec 3

This document discusses the principle of inclusion and exclusion as it relates to set theory and probability. It begins by explaining the principle for two sets A and B, that the size of the union A ∪ B equals the size of A plus the size of B minus the size of their intersection A ∩ B. It then generalizes this to work for any number of sets. Examples are provided to demonstrate calculating sizes of unions, intersections, and complements of sets. The pigeonhole principle is also introduced, which states that if more items are placed in bins than the number of bins, one bin must contain multiple items.

Introduction to set theory

This document provides an introduction to the key concepts of set theory, including:
- A set is a well-defined collection of objects or elements. Sets can be defined by listing elements or describing membership rules.
- Common notations are presented for defining sets, elements, membership, subsets, unions, intersections, complements, and cardinality.
- Finite and infinite sets are discussed. Special sets like the empty set, power set, and universal set are introduced.
- Venn diagrams are used to visually represent relationships between sets such as subsets, unions, intersections, and complements.

Discrete Structure Lecture #7 & 8.pdf

The document discusses basic concepts of set theory including defining sets, set operations like union and intersection, proofs of set identities using techniques like mutual inclusion, and applications of set theory concepts like the pigeonhole principle. It provides examples and explanations of fundamental set theory terms and concepts such as members, subsets, power sets, Venn diagrams, and proofs of set properties and identities.

Section3 2

The document discusses various principles of combinatorics including the multiplication principle, addition principle, principle of inclusion and exclusion, and pigeonhole principle. It provides examples and explanations of how to use these principles to calculate the number of possible outcomes for different combinatorial problems, such as determining the number of possible rolls of dice, ways to choose a committee from a group of people, or number of possible outfits given different articles of clothing.

Set Operations

The document discusses set operations including union, intersection, difference, complement, and disjoint sets. It provides formal definitions and examples for each operation. Properties of the various operations are listed, such as the commutative, associative, identity, and domination laws. Methods for proving set identities are also described.

The principle of inclusion and exclusion for three sets by sharvari

The principle of inclusion and exclusion for three sets by sharvari

2.2 Set Operations

2.2 Set Operations

Pdm presentation

Pdm presentation

mathematical sets.pdf

mathematical sets.pdf

8-Sets-2.ppt

8-Sets-2.ppt

20200911-XI-Maths-Sets-2 of 2-Ppt.pdf

20200911-XI-Maths-Sets-2 of 2-Ppt.pdf

2 》set operation.pdf

2 》set operation.pdf

CPSC 125 Ch 3 Sec 3

CPSC 125 Ch 3 Sec 3

Introduction to set theory

Introduction to set theory

Discrete Structure Lecture #7 & 8.pdf

Discrete Structure Lecture #7 & 8.pdf

Section3 2

Section3 2

Set Operations

Set Operations

Your One-Stop Shop for Python Success: Top 10 US Python Development Providers

Simplify your search for a reliable Python development partner! This list presents the top 10 trusted US providers offering comprehensive Python development services, ensuring your project's success from conception to completion.

A Comprehensive Guide to DeFi Development Services in 2024

DeFi represents a paradigm shift in the financial industry. Instead of relying on traditional, centralized institutions like banks, DeFi leverages blockchain technology to create a decentralized network of financial services. This means that financial transactions can occur directly between parties, without intermediaries, using smart contracts on platforms like Ethereum.
In 2024, we are witnessing an explosion of new DeFi projects and protocols, each pushing the boundaries of what’s possible in finance.
In summary, DeFi in 2024 is not just a trend; it’s a revolution that democratizes finance, enhances security and transparency, and fosters continuous innovation. As we proceed through this presentation, we'll explore the various components and services of DeFi in detail, shedding light on how they are transforming the financial landscape.
At Intelisync, we specialize in providing comprehensive DeFi development services tailored to meet the unique needs of our clients. From smart contract development to dApp creation and security audits, we ensure that your DeFi project is built with innovation, security, and scalability in mind. Trust Intelisync to guide you through the intricate landscape of decentralized finance and unlock the full potential of blockchain technology.
Ready to take your DeFi project to the next level? Partner with Intelisync for expert DeFi development services today!

Recommendation System using RAG Architecture

Concept of how to create a RAG arhcitecture

Artificial Intelligence for XMLDevelopment

In the rapidly evolving landscape of technologies, XML continues to play a vital role in structuring, storing, and transporting data across diverse systems. The recent advancements in artificial intelligence (AI) present new methodologies for enhancing XML development workflows, introducing efficiency, automation, and intelligent capabilities. This presentation will outline the scope and perspective of utilizing AI in XML development. The potential benefits and the possible pitfalls will be highlighted, providing a balanced view of the subject.
We will explore the capabilities of AI in understanding XML markup languages and autonomously creating structured XML content. Additionally, we will examine the capacity of AI to enrich plain text with appropriate XML markup. Practical examples and methodological guidelines will be provided to elucidate how AI can be effectively prompted to interpret and generate accurate XML markup.
Further emphasis will be placed on the role of AI in developing XSLT, or schemas such as XSD and Schematron. We will address the techniques and strategies adopted to create prompts for generating code, explaining code, or refactoring the code, and the results achieved.
The discussion will extend to how AI can be used to transform XML content. In particular, the focus will be on the use of AI XPath extension functions in XSLT, Schematron, Schematron Quick Fixes, or for XML content refactoring.
The presentation aims to deliver a comprehensive overview of AI usage in XML development, providing attendees with the necessary knowledge to make informed decisions. Whether you’re at the early stages of adopting AI or considering integrating it in advanced XML development, this presentation will cover all levels of expertise.
By highlighting the potential advantages and challenges of integrating AI with XML development tools and languages, the presentation seeks to inspire thoughtful conversation around the future of XML development. We’ll not only delve into the technical aspects of AI-powered XML development but also discuss practical implications and possible future directions.

Serial Arm Control in Real Time Presentation

Serial Arm Control in Real Time

Operating System Used by Users in day-to-day life.pptx

Dive into the realm of operating systems (OS) with Pravash Chandra Das, a seasoned Digital Forensic Analyst, as your guide. 🚀 This comprehensive presentation illuminates the core concepts, types, and evolution of OS, essential for understanding modern computing landscapes.
Beginning with the foundational definition, Das clarifies the pivotal role of OS as system software orchestrating hardware resources, software applications, and user interactions. Through succinct descriptions, he delineates the diverse types of OS, from single-user, single-task environments like early MS-DOS iterations, to multi-user, multi-tasking systems exemplified by modern Linux distributions.
Crucial components like the kernel and shell are dissected, highlighting their indispensable functions in resource management and user interface interaction. Das elucidates how the kernel acts as the central nervous system, orchestrating process scheduling, memory allocation, and device management. Meanwhile, the shell serves as the gateway for user commands, bridging the gap between human input and machine execution. 💻
The narrative then shifts to a captivating exploration of prominent desktop OSs, Windows, macOS, and Linux. Windows, with its globally ubiquitous presence and user-friendly interface, emerges as a cornerstone in personal computing history. macOS, lauded for its sleek design and seamless integration with Apple's ecosystem, stands as a beacon of stability and creativity. Linux, an open-source marvel, offers unparalleled flexibility and security, revolutionizing the computing landscape. 🖥️
Moving to the realm of mobile devices, Das unravels the dominance of Android and iOS. Android's open-source ethos fosters a vibrant ecosystem of customization and innovation, while iOS boasts a seamless user experience and robust security infrastructure. Meanwhile, discontinued platforms like Symbian and Palm OS evoke nostalgia for their pioneering roles in the smartphone revolution.
The journey concludes with a reflection on the ever-evolving landscape of OS, underscored by the emergence of real-time operating systems (RTOS) and the persistent quest for innovation and efficiency. As technology continues to shape our world, understanding the foundations and evolution of operating systems remains paramount. Join Pravash Chandra Das on this illuminating journey through the heart of computing. 🌟

Unlock the Future of Search with MongoDB Atlas_ Vector Search Unleashed.pdf

Discover how MongoDB Atlas and vector search technology can revolutionize your application's search capabilities. This comprehensive presentation covers:
* What is Vector Search?
* Importance and benefits of vector search
* Practical use cases across various industries
* Step-by-step implementation guide
* Live demos with code snippets
* Enhancing LLM capabilities with vector search
* Best practices and optimization strategies
Perfect for developers, AI enthusiasts, and tech leaders. Learn how to leverage MongoDB Atlas to deliver highly relevant, context-aware search results, transforming your data retrieval process. Stay ahead in tech innovation and maximize the potential of your applications.
#MongoDB #VectorSearch #AI #SemanticSearch #TechInnovation #DataScience #LLM #MachineLearning #SearchTechnology

Programming Foundation Models with DSPy - Meetup Slides

Prompting language models is hard, while programming language models is easy. In this talk, I will discuss the state-of-the-art framework DSPy for programming foundation models with its powerful optimizers and runtime constraint system.

HCL Notes and Domino License Cost Reduction in the World of DLAU

Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-and-domino-license-cost-reduction-in-the-world-of-dlau/
The introduction of DLAU and the CCB & CCX licensing model caused quite a stir in the HCL community. As a Notes and Domino customer, you may have faced challenges with unexpected user counts and license costs. You probably have questions on how this new licensing approach works and how to benefit from it. Most importantly, you likely have budget constraints and want to save money where possible. Don’t worry, we can help with all of this!
We’ll show you how to fix common misconfigurations that cause higher-than-expected user counts, and how to identify accounts which you can deactivate to save money. There are also frequent patterns that can cause unnecessary cost, like using a person document instead of a mail-in for shared mailboxes. We’ll provide examples and solutions for those as well. And naturally we’ll explain the new licensing model.
Join HCL Ambassador Marc Thomas in this webinar with a special guest appearance from Franz Walder. It will give you the tools and know-how to stay on top of what is going on with Domino licensing. You will be able lower your cost through an optimized configuration and keep it low going forward.
These topics will be covered
- Reducing license cost by finding and fixing misconfigurations and superfluous accounts
- How do CCB and CCX licenses really work?
- Understanding the DLAU tool and how to best utilize it
- Tips for common problem areas, like team mailboxes, functional/test users, etc
- Practical examples and best practices to implement right away

Driving Business Innovation: Latest Generative AI Advancements & Success Story

Are you ready to revolutionize how you handle data? Join us for a webinar where we’ll bring you up to speed with the latest advancements in Generative AI technology and discover how leveraging FME with tools from giants like Google Gemini, Amazon, and Microsoft OpenAI can supercharge your workflow efficiency.
During the hour, we’ll take you through:
Guest Speaker Segment with Hannah Barrington: Dive into the world of dynamic real estate marketing with Hannah, the Marketing Manager at Workspace Group. Hear firsthand how their team generates engaging descriptions for thousands of office units by integrating diverse data sources—from PDF floorplans to web pages—using FME transformers, like OpenAIVisionConnector and AnthropicVisionConnector. This use case will show you how GenAI can streamline content creation for marketing across the board.
Ollama Use Case: Learn how Scenario Specialist Dmitri Bagh has utilized Ollama within FME to input data, create custom models, and enhance security protocols. This segment will include demos to illustrate the full capabilities of FME in AI-driven processes.
Custom AI Models: Discover how to leverage FME to build personalized AI models using your data. Whether it’s populating a model with local data for added security or integrating public AI tools, find out how FME facilitates a versatile and secure approach to AI.
We’ll wrap up with a live Q&A session where you can engage with our experts on your specific use cases, and learn more about optimizing your data workflows with AI.
This webinar is ideal for professionals seeking to harness the power of AI within their data management systems while ensuring high levels of customization and security. Whether you're a novice or an expert, gain actionable insights and strategies to elevate your data processes. Join us to see how FME and AI can revolutionize how you work with data!

Generating privacy-protected synthetic data using Secludy and Milvus

During this demo, the founders of Secludy will demonstrate how their system utilizes Milvus to store and manipulate embeddings for generating privacy-protected synthetic data. Their approach not only maintains the confidentiality of the original data but also enhances the utility and scalability of LLMs under privacy constraints. Attendees, including machine learning engineers, data scientists, and data managers, will witness first-hand how Secludy's integration with Milvus empowers organizations to harness the power of LLMs securely and efficiently.

GraphRAG for Life Science to increase LLM accuracy

GraphRAG for life science domain, where you retriever information from biomedical knowledge graphs using LLMs to increase the accuracy and performance of generated answers

AWS Cloud Cost Optimization Presentation.pptx

This presentation provides valuable insights into effective cost-saving techniques on AWS. Learn how to optimize your AWS resources by rightsizing, increasing elasticity, picking the right storage class, and choosing the best pricing model. Additionally, discover essential governance mechanisms to ensure continuous cost efficiency. Whether you are new to AWS or an experienced user, this presentation provides clear and practical tips to help you reduce your cloud costs and get the most out of your budget.

Monitoring and Managing Anomaly Detection on OpenShift.pdf

Monitoring and Managing Anomaly Detection on OpenShift
Overview
Dive into the world of anomaly detection on edge devices with our comprehensive hands-on tutorial. This SlideShare presentation will guide you through the entire process, from data collection and model training to edge deployment and real-time monitoring. Perfect for those looking to implement robust anomaly detection systems on resource-constrained IoT/edge devices.
Key Topics Covered
1. Introduction to Anomaly Detection
- Understand the fundamentals of anomaly detection and its importance in identifying unusual behavior or failures in systems.
2. Understanding Edge (IoT)
- Learn about edge computing and IoT, and how they enable real-time data processing and decision-making at the source.
3. What is ArgoCD?
- Discover ArgoCD, a declarative, GitOps continuous delivery tool for Kubernetes, and its role in deploying applications on edge devices.
4. Deployment Using ArgoCD for Edge Devices
- Step-by-step guide on deploying anomaly detection models on edge devices using ArgoCD.
5. Introduction to Apache Kafka and S3
- Explore Apache Kafka for real-time data streaming and Amazon S3 for scalable storage solutions.
6. Viewing Kafka Messages in the Data Lake
- Learn how to view and analyze Kafka messages stored in a data lake for better insights.
7. What is Prometheus?
- Get to know Prometheus, an open-source monitoring and alerting toolkit, and its application in monitoring edge devices.
8. Monitoring Application Metrics with Prometheus
- Detailed instructions on setting up Prometheus to monitor the performance and health of your anomaly detection system.
9. What is Camel K?
- Introduction to Camel K, a lightweight integration framework built on Apache Camel, designed for Kubernetes.
10. Configuring Camel K Integrations for Data Pipelines
- Learn how to configure Camel K for seamless data pipeline integrations in your anomaly detection workflow.
11. What is a Jupyter Notebook?
- Overview of Jupyter Notebooks, an open-source web application for creating and sharing documents with live code, equations, visualizations, and narrative text.
12. Jupyter Notebooks with Code Examples
- Hands-on examples and code snippets in Jupyter Notebooks to help you implement and test anomaly detection models.

WeTestAthens: Postman's AI & Automation Techniques

Postman's AI and Automation Techniques

TrustArc Webinar - 2024 Global Privacy Survey

How does your privacy program stack up against your peers? What challenges are privacy teams tackling and prioritizing in 2024?
In the fifth annual Global Privacy Benchmarks Survey, we asked over 1,800 global privacy professionals and business executives to share their perspectives on the current state of privacy inside and outside of their organizations. This year’s report focused on emerging areas of importance for privacy and compliance professionals, including considerations and implications of Artificial Intelligence (AI) technologies, building brand trust, and different approaches for achieving higher privacy competence scores.
See how organizational priorities and strategic approaches to data security and privacy are evolving around the globe.
This webinar will review:
- The top 10 privacy insights from the fifth annual Global Privacy Benchmarks Survey
- The top challenges for privacy leaders, practitioners, and organizations in 2024
- Key themes to consider in developing and maintaining your privacy program

Energy Efficient Video Encoding for Cloud and Edge Computing Instances

Energy Efficient Video Encoding for Cloud and Edge Computing Instances

UI5 Controls simplified - UI5con2024 presentation

UI5con 2024 presentation

Taking AI to the Next Level in Manufacturing.pdf

Read Taking AI to the Next Level in Manufacturing to gain insights on AI adoption in the manufacturing industry, such as:
1. How quickly AI is being implemented in manufacturing.
2. Which barriers stand in the way of AI adoption.
3. How data quality and governance form the backbone of AI.
4. Organizational processes and structures that may inhibit effective AI adoption.
6. Ideas and approaches to help build your organization's AI strategy.

Your One-Stop Shop for Python Success: Top 10 US Python Development Providers

Your One-Stop Shop for Python Success: Top 10 US Python Development Providers

Overcoming the PLG Trap: Lessons from Canva's Head of Sales & Head of EMEA Da...

Overcoming the PLG Trap: Lessons from Canva's Head of Sales & Head of EMEA Da...

A Comprehensive Guide to DeFi Development Services in 2024

A Comprehensive Guide to DeFi Development Services in 2024

Recommendation System using RAG Architecture

Recommendation System using RAG Architecture

Artificial Intelligence for XMLDevelopment

Artificial Intelligence for XMLDevelopment

Serial Arm Control in Real Time Presentation

Serial Arm Control in Real Time Presentation

Operating System Used by Users in day-to-day life.pptx

Operating System Used by Users in day-to-day life.pptx

Unlock the Future of Search with MongoDB Atlas_ Vector Search Unleashed.pdf

Unlock the Future of Search with MongoDB Atlas_ Vector Search Unleashed.pdf

Programming Foundation Models with DSPy - Meetup Slides

Programming Foundation Models with DSPy - Meetup Slides

HCL Notes and Domino License Cost Reduction in the World of DLAU

HCL Notes and Domino License Cost Reduction in the World of DLAU

Driving Business Innovation: Latest Generative AI Advancements & Success Story

Driving Business Innovation: Latest Generative AI Advancements & Success Story

Generating privacy-protected synthetic data using Secludy and Milvus

Generating privacy-protected synthetic data using Secludy and Milvus

GraphRAG for Life Science to increase LLM accuracy

GraphRAG for Life Science to increase LLM accuracy

AWS Cloud Cost Optimization Presentation.pptx

AWS Cloud Cost Optimization Presentation.pptx

Monitoring and Managing Anomaly Detection on OpenShift.pdf

Monitoring and Managing Anomaly Detection on OpenShift.pdf

WeTestAthens: Postman's AI & Automation Techniques

WeTestAthens: Postman's AI & Automation Techniques

TrustArc Webinar - 2024 Global Privacy Survey

TrustArc Webinar - 2024 Global Privacy Survey

Energy Efficient Video Encoding for Cloud and Edge Computing Instances

Energy Efficient Video Encoding for Cloud and Edge Computing Instances

UI5 Controls simplified - UI5con2024 presentation

UI5 Controls simplified - UI5con2024 presentation

Taking AI to the Next Level in Manufacturing.pdf

Taking AI to the Next Level in Manufacturing.pdf

- 1. Introduction to Combinatorics A.Benedict Balbuena Institute of Mathematics, University of the Philippines in Diliman 11.1.2008 A.B.C.Balbuena (UP-Math) Introduction to Combinatorics 11.1.2008 1 / 10
- 2. Addition Rule Theorem If A1 , A2 , ..., An are disjoint sets (n < ∞, n ∈ N) then: |A1 ∪ A2 ∪ ... ∪ An | = |A1 | + |A2 | + ... + |An | Works only for disjoint sets One seat in a presidential working commitee is reserved for either a senator or a party-list representative. How many possible choices are there for the seat if there are 23 senators and 27 party-list representatives? A.B.C.Balbuena (UP-Math) Introduction to Combinatorics 11.1.2008 2 / 10
- 3. Addition Rule Theorem If A1 , A2 , ..., An are disjoint sets (n < ∞, n ∈ N) then: |A1 ∪ A2 ∪ ... ∪ An | = |A1 | + |A2 | + ... + |An | Works only for disjoint sets One seat in a presidential working commitee is reserved for either a senator or a party-list representative. How many possible choices are there for the seat if there are 23 senators and 27 party-list representatives? A.B.C.Balbuena (UP-Math) Introduction to Combinatorics 11.1.2008 2 / 10
- 4. Addition Rule Theorem If A1 , A2 , ..., An are disjoint sets (n < ∞, n ∈ N) then: |A1 ∪ A2 ∪ ... ∪ An | = |A1 | + |A2 | + ... + |An | Works only for disjoint sets One seat in a presidential working commitee is reserved for either a senator or a party-list representative. How many possible choices are there for the seat if there are 23 senators and 27 party-list representatives? A.B.C.Balbuena (UP-Math) Introduction to Combinatorics 11.1.2008 2 / 10
- 5. Product Rule Recall: A × B = {(a, b)|a ∈ A, b ∈ B} Theorem If A1 , A2 , ..., An are sets (n < ∞, n ∈ N) then: |A1 × A2 × ... × An | = |A1 ||A2 |...|An | Interpret as number of ways to pick one item from A1 , one item from A2 , ..., one item from An A.B.C.Balbuena (UP-Math) Introduction to Combinatorics 11.1.2008 3 / 10
- 6. Product Rule Recall: A × B = {(a, b)|a ∈ A, b ∈ B} Theorem If A1 , A2 , ..., An are sets (n < ∞, n ∈ N) then: |A1 × A2 × ... × An | = |A1 ||A2 |...|An | Interpret as number of ways to pick one item from A1 , one item from A2 , ..., one item from An A.B.C.Balbuena (UP-Math) Introduction to Combinatorics 11.1.2008 3 / 10
- 7. Product Rule Recall: A × B = {(a, b)|a ∈ A, b ∈ B} Theorem If A1 , A2 , ..., An are sets (n < ∞, n ∈ N) then: |A1 × A2 × ... × An | = |A1 ||A2 |...|An | Interpret as number of ways to pick one item from A1 , one item from A2 , ..., one item from An A.B.C.Balbuena (UP-Math) Introduction to Combinatorics 11.1.2008 3 / 10
- 8. Examples 1 How many bit-strings are there of length n? 2 How many functions are there from a set of m elements to one with n elements? 3 How many possible mobile phone numbers are there in the Philippines? A.B.C.Balbuena (UP-Math) Introduction to Combinatorics 11.1.2008 4 / 10
- 9. Inclusion-Exclusion Principle Theorem Let A, B be sets. Then: |A ∪ B| = |A| + |B| − |A ∩ B| Proof. By deﬁnitions of set difference and intersection, B = (A ∩ B) ∪ (B − A).Now, A ∩ B and B − A are disjoint sets.By the previous thm, |B| = |A ∩ B| + |B − A|.Since A and B − A are disjoint, |A ∪ B| = |A| + |B − A|.Replacing B − A, we have |A ∪ B| = |A| + |B| − |A ∩ B| How many bit strings of length of length 8 start with a 1 or end with 00? A.B.C.Balbuena (UP-Math) Introduction to Combinatorics 11.1.2008 5 / 10
- 10. Inclusion-Exclusion Principle Theorem Let A, B be sets. Then: |A ∪ B| = |A| + |B| − |A ∩ B| Proof. By deﬁnitions of set difference and intersection, B = (A ∩ B) ∪ (B − A).Now, A ∩ B and B − A are disjoint sets.By the previous thm, |B| = |A ∩ B| + |B − A|.Since A and B − A are disjoint, |A ∪ B| = |A| + |B − A|.Replacing B − A, we have |A ∪ B| = |A| + |B| − |A ∩ B| How many bit strings of length of length 8 start with a 1 or end with 00? A.B.C.Balbuena (UP-Math) Introduction to Combinatorics 11.1.2008 5 / 10
- 11. Inclusion-Exclusion Principle Theorem Let A, B be sets. Then: |A ∪ B| = |A| + |B| − |A ∩ B| Proof. By deﬁnitions of set difference and intersection, B = (A ∩ B) ∪ (B − A).Now, A ∩ B and B − A are disjoint sets.By the previous thm, |B| = |A ∩ B| + |B − A|.Since A and B − A are disjoint, |A ∪ B| = |A| + |B − A|.Replacing B − A, we have |A ∪ B| = |A| + |B| − |A ∩ B| How many bit strings of length of length 8 start with a 1 or end with 00? A.B.C.Balbuena (UP-Math) Introduction to Combinatorics 11.1.2008 5 / 10
- 12. Inclusion-Exclusion Principle Theorem Let A, B be sets. Then: |A ∪ B| = |A| + |B| − |A ∩ B| Proof. By deﬁnitions of set difference and intersection, B = (A ∩ B) ∪ (B − A).Now, A ∩ B and B − A are disjoint sets.By the previous thm, |B| = |A ∩ B| + |B − A|.Since A and B − A are disjoint, |A ∪ B| = |A| + |B − A|.Replacing B − A, we have |A ∪ B| = |A| + |B| − |A ∩ B| How many bit strings of length of length 8 start with a 1 or end with 00? A.B.C.Balbuena (UP-Math) Introduction to Combinatorics 11.1.2008 5 / 10
- 13. Inclusion-Exclusion Principle Theorem Let A, B be sets. Then: |A ∪ B| = |A| + |B| − |A ∩ B| Proof. By deﬁnitions of set difference and intersection, B = (A ∩ B) ∪ (B − A).Now, A ∩ B and B − A are disjoint sets.By the previous thm, |B| = |A ∩ B| + |B − A|.Since A and B − A are disjoint, |A ∪ B| = |A| + |B − A|.Replacing B − A, we have |A ∪ B| = |A| + |B| − |A ∩ B| How many bit strings of length of length 8 start with a 1 or end with 00? A.B.C.Balbuena (UP-Math) Introduction to Combinatorics 11.1.2008 5 / 10
- 14. Inclusion-Exclusion Principle Theorem Let A, B be sets. Then: |A ∪ B| = |A| + |B| − |A ∩ B| Proof. By deﬁnitions of set difference and intersection, B = (A ∩ B) ∪ (B − A).Now, A ∩ B and B − A are disjoint sets.By the previous thm, |B| = |A ∩ B| + |B − A|.Since A and B − A are disjoint, |A ∪ B| = |A| + |B − A|.Replacing B − A, we have |A ∪ B| = |A| + |B| − |A ∩ B| How many bit strings of length of length 8 start with a 1 or end with 00? A.B.C.Balbuena (UP-Math) Introduction to Combinatorics 11.1.2008 5 / 10
- 15. A password on a social networking site is six characters long, where each character is a letter or a digit. Each password must contain at least one digit. How many passwords are there? What if a password is six to eight chracters long? A.B.C.Balbuena (UP-Math) Introduction to Combinatorics 11.1.2008 6 / 10
- 16. Suppose we have 3 blue shirts, 2 red shirts, and 1 green shirt. We also have 2 gray pants and 3 brown pants. How many outﬁts are possible? (Two pieces of clothing with the same color are still considered distinct; assume that they have slightly different shades.) S := set of shirts P := set of pants. We form an outﬁt by picking one shirt and one pair of pants. By the Product Rule, there are (6)(5) = 30 outﬁts. A.B.C.Balbuena (UP-Math) Introduction to Combinatorics 11.1.2008 7 / 10
- 17. Suppose we have 3 blue shirts, 2 red shirts, and 1 green shirt. We also have 2 gray pants and 3 brown pants. How many outﬁts are possible? (Two pieces of clothing with the same color are still considered distinct; assume that they have slightly different shades.) S := set of shirts P := set of pants. We form an outﬁt by picking one shirt and one pair of pants. By the Product Rule, there are (6)(5) = 30 outﬁts. A.B.C.Balbuena (UP-Math) Introduction to Combinatorics 11.1.2008 7 / 10
- 18. What if gray pants go with only blue and red shirts and brown pants go with only green and red shirts. How many matching outﬁts are there? Count the number of matching outﬁts by counting the number of mismatching outﬁts. By the Product Rule, there are (2)(1) = 2 mismatching gray-green outﬁts and (3)(3) = 9 mismatching brown-blue outﬁts. Therefore, there are 3029 = 19 matching outﬁts. A.B.C.Balbuena (UP-Math) Introduction to Combinatorics 11.1.2008 8 / 10
- 19. How many ways can the ﬁve game NBA Finals Series be decided? (The series is decided when a team has three wins or three losses) A.B.C.Balbuena (UP-Math) Introduction to Combinatorics 11.1.2008 9 / 10
- 20. A.B.C.Balbuena (UP-Math) Introduction to Combinatorics 11.1.2008 10 / 10