6.1 Partitions
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Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.
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6.3 Equivalences versus partitions
This document introduces equivalence relations and partitions. It defines an equivalence relation as a binary relation that is reflexive, symmetric, and transitive. Equivalence relations partition a set into disjoint equivalence classes that cover the entire set. The quotient set of a set by an equivalence relation consists of the equivalence classes. Every equivalence relation determines a partition, and every partition determines an equivalence relation. Examples are provided to illustrate these concepts using the equivalence relation of congruence modulo 3 on the integers.
6.2 Reflexivity, symmetry and transitivity (handout)
Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.
6.2 Reflexivity, symmetry and transitivity (dynamic slides)
This document introduces basic concepts of set theory including definitions of reflexive, symmetric and transitive relations. It provides examples of relations between lines that are parallel or perpendicular. It also discusses relations between people like siblings and ancestors. The document proves properties of congruence relations and discusses counting relations with certain properties over sets with a given number of elements. It addresses a claim about relations that are symmetric and transitive necessarily being reflexive, finding a counterexample.
Axiom of Choice
This document discusses the axiom of choice in set theory. It provides definitions of key terms like well-ordering, partial ordering, and Zorn's lemma. It also covers some equivalents and consequences of the axiom of choice, including the well-ordering principle and Banach-Tarski paradox. The axiom of choice allows choosing one element from each nonempty set in a collection of disjoint sets and guarantees the existence of a choice set.
On Review of the Cluster Point of a Set in a Topological Space
This document summarizes key concepts in topology related to the cluster point or limit point of a set in a topological space. It defines a cluster/limit point as a point such that every neighborhood of it intersects the set in at least one other point. Theorems establish that a set and its cluster points together form its closure, and that a set is closed if it contains all its cluster points. No finite set can have a cluster point. The union of a set's cluster points with the set of another's forms the cluster points of their union.
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.
L04 (1)
We will define what is a function “formally”, and then
in the next lecture we will use this concept in counting.
We will also study the pigeonhole principle and its applications
Discrete Math Lecture 02: First Order Logic
Last time we talked about propositional logic, a logic on simple statements.
This time we will talk about first order logic, a logic on quantified statements.
First order logic is much more expressive than propositional logic.
The topics on first order logic are:
1-Quantifiers
2-Negation
3-Multiple quantifiers
4-Arguments of quantified statements
Recommended
6.3 Equivalences versus partitions
This document introduces equivalence relations and partitions. It defines an equivalence relation as a binary relation that is reflexive, symmetric, and transitive. Equivalence relations partition a set into disjoint equivalence classes that cover the entire set. The quotient set of a set by an equivalence relation consists of the equivalence classes. Every equivalence relation determines a partition, and every partition determines an equivalence relation. Examples are provided to illustrate these concepts using the equivalence relation of congruence modulo 3 on the integers.
6.2 Reflexivity, symmetry and transitivity (handout)
Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.
6.2 Reflexivity, symmetry and transitivity (dynamic slides)
This document introduces basic concepts of set theory including definitions of reflexive, symmetric and transitive relations. It provides examples of relations between lines that are parallel or perpendicular. It also discusses relations between people like siblings and ancestors. The document proves properties of congruence relations and discusses counting relations with certain properties over sets with a given number of elements. It addresses a claim about relations that are symmetric and transitive necessarily being reflexive, finding a counterexample.
Axiom of Choice
This document discusses the axiom of choice in set theory. It provides definitions of key terms like well-ordering, partial ordering, and Zorn's lemma. It also covers some equivalents and consequences of the axiom of choice, including the well-ordering principle and Banach-Tarski paradox. The axiom of choice allows choosing one element from each nonempty set in a collection of disjoint sets and guarantees the existence of a choice set.
On Review of the Cluster Point of a Set in a Topological Space
This document summarizes key concepts in topology related to the cluster point or limit point of a set in a topological space. It defines a cluster/limit point as a point such that every neighborhood of it intersects the set in at least one other point. Theorems establish that a set and its cluster points together form its closure, and that a set is closed if it contains all its cluster points. No finite set can have a cluster point. The union of a set's cluster points with the set of another's forms the cluster points of their union.
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.
L04 (1)
We will define what is a function “formally”, and then
in the next lecture we will use this concept in counting.
We will also study the pigeonhole principle and its applications
Discrete Math Lecture 02: First Order Logic
Last time we talked about propositional logic, a logic on simple statements.
This time we will talk about first order logic, a logic on quantified statements.
First order logic is much more expressive than propositional logic.
The topics on first order logic are:
1-Quantifiers
2-Negation
3-Multiple quantifiers
4-Arguments of quantified statements
Limits BY ATC
ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
VISIT US @
www.anuragtyagiclasses.com
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The document discusses the pigeonhole principle, which states that if n objects are put into m containers where n > m, then at least one container must contain more than one object. It provides various formulations and applications of the principle in areas like data compression, hash tables, and the Chinese Remainder Theorem. The history of the principle is traced back to Dirichlet, who described it as the "drawer principle" or "shelf principle" in 1834. Examples are given for problems involving birthdays, friend relationships, and geometry that can be solved using the pigeonhole principle.
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Limits BY ATC
ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
VISIT US @
www.anuragtyagiclasses.com
Raices primitivas
This document introduces the concept of order of an element modulo n and uses it to prove theorems about when an integer n satisfies n^2 + 1 or more generally satisfies a cyclotomic polynomial modulo a prime p. It begins by stating and proving the n^2 + 1 lemma, which says a prime p satisfies p | n^2 + 1 if and only if p ≡ 1 (mod 4). It introduces the concepts of order, primitive roots, and cyclotomic polynomials to generalize this result. It concludes by stating and proving a theorem about when a cyclotomic polynomial of an integer a is divisible by a prime p.
Binary relations
The document defines binary relations and provides examples of binary relations between sets. It then discusses properties of binary relations such as being reflexive, symmetric, transitive, complete, antisymmetric, asymmetric, or irreflexive. It introduces the concepts of preorders, orders, equivalence relations, and partitions. A preorder is a binary relation that is transitive and either reflexive or irreflexive. An order is a complete, transitive, and antisymmetric preorder. An equivalence relation is a reflexive, symmetric, and transitive binary relation that partitions a set into equivalence classes. Utility functions are introduced as a way to represent preorders, where a utility function u represents a preorder R if xRy if and
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This document discusses separating shuffle regular expressions (SSRE) for describing languages of data words. SSRE extend regular expressions with a separating shuffle operation. The document defines SSRE and proves several results about their expressiveness and decidability properties in comparison to register automata, data automata, and first-order logic on data words. Key results include: 1) every data automata definable language is definable by a SSRE with homomorphisms; 2) the emptiness problem for SSRE with homomorphisms is undecidable; and 3) there are languages definable by SSRE that cannot be defined by register automata. The document raises several open questions and conjectures about SSRE and their relationships to other
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This document provides an overview of the pigeonhole principle from discrete mathematics. It defines the principle as stating that if n items are put into m containers, with n > m, then at least one container must contain more than one item. It then provides several examples to illustrate applications of the principle, such as hand shaking, hair counting, and birthday problems. It also presents some alternative formulations of the principle and lists references for further reading.
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This document provides an overview of key concepts in set theory, including:
1) Sets can be defined by listing elements or using predicates, and basic set operations include membership, equality, subsets, and power sets.
2) Relationships between sets such as subsets, supersets, proper subsets are defined, and examples are given to illustrate concepts like open and closed intervals.
3) Common set notations are introduced for natural numbers, integers, rational numbers, and real numbers. Binary operations on sets are defined to be well-defined and keep the set closed under the operation.
Propositional Logic and Pridicate logic
This document provides an introduction to logic, including propositional logic and predicate calculus. It defines key concepts such as logical values, propositions, operators, truth tables, logical expressions, worlds, models, inference rules, quantification, and definitions. Propositional logic manipulates true and false values using operators like AND and OR. Predicate calculus extends this to allow predicates, constants, functions, and quantification over variables. Inference involves applying rules to derive new statements, but the search space grows too large to feasibly perform by hand.
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The document discusses propositional logic and covers topics like propositional variables, truth tables, logical equivalence, predicates, and quantifiers. It defines key concepts such as propositions, tautologies, contradictions, predicates, universal and existential quantifiers. Examples are provided to illustrate different types of truth tables, logical equivalences like De Morgan's laws, and uses of quantifiers.
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This paper investigates sufficient conditions for the multiplication operator Mz to be reflexive on weighted Hardy spaces Lp(β). It proves that if Lp(β) = Lp∞(β), meaning the space of multipliers is equal to the whole space, then Mz is reflexive on Lp(β). The proof shows that in this case, the algebra of operators commuting with Mz is equal to the smallest subalgebra containing Mz and closed in the weak operator topology, which is the definition of a reflexive operator. This establishes reflexivity of the multiplication operator under the given condition on the weighted Hardy space.
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The document defines and provides examples of rings and ideals. Some key points:
- A ring consists of a set with operations of addition and multiplication satisfying certain properties like commutativity and associativity.
- Common examples of rings include the integers, rational numbers, real numbers, polynomials, and matrices.
- An ideal is a subset of a ring that is closed under addition and multiplication. Ideals play an important role in ring theory.
- Given an ideal I of a ring R, a quotient ring R/I can be constructed by identifying elements of R that differ by an element of I. Operations are defined on these equivalence classes.
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Selected items from the introduction to set theory and to methodology and philosophy of mathematics and computer programming.
5.8 Permutations (dynamic slides)
This document provides an introduction to permutations. It begins by defining a permutation as a bijection from a set to itself. It then gives examples of permutations of small sets. The document proves several properties of permutations, including that the number of permutations of a set with n elements is n!. It introduces the concept of powers of permutations and proves related properties. It concludes by proving additional properties of permutations using an algebraic approach that relies on basic properties like associativity of composition.
5.7 Function powers
Selected items from set theory and from methodology and philosophy of mathematics and computer programming.
5.6 Function inverse. A handout.
Selected items from set theory and methodology and philosophy of mathematics and computer programming.
5.6 Function inverse. Dynamic slides.
Selected items from set theory and methodology and philosophy of mathematics and computer programming.
5.5 Injective and surjective functions. A handout.
Selected items from set theory and methodology and philosophy of mathematics and computer programming.
5.5 Injective and surjective functions. Dynamic slides.
Selected items from set theory and from methodology and philosophy of mathematics and computer programming.
5.1 Defining and visualizing functions. A handout.
This document introduces concepts related to functions including:
- Defining functions in terms of unique mappings between inputs and outputs
- Distinguishing between total, partial, and non-functions
- Specifying domains and ranges
- Using vertical line tests to identify functions from graphs
- Examples of functions defined by formulas or mappings
5.1 Defining and visualizing functions. Dynamic slides.
Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.
1.8 Separation schema
Selected items from the Introduction to Set Theory and to Methodology and Philosophy of Mathematics and Computer Programming.
1.4 Abstract objects and expressions
Selected items from the Introduction to Set Theory and to Methodology and Philosophy of Mathematics and Computer Programming.
1.2 Axiom of pair
Selected items from the Introduction to Set Theory and to Methodology and Philosophy of Mathematics and Computer Programming.
1.11 Mathematical induction
Selected items from Introduction to Set Theory and Methodology and Philosophy of Mathematics and Computer Programming.
1.7 The sets of numbers
Selected items from Introduction to Set Theory and Methodology and Philosophy of Mathematics and Computer Programming.
1.6 Subsets
Selected items from Introduction to Set Theory and Methodology and Philosophy of Mathematics and Computer Programming.
1.1 Notions of set and membership
Selected items from Introduction to Set Theory and Methodology and Philosophy of Mathematics and Computer Programming.
5.2. Function composition
Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programing.
4.7 Powers of binary relations
Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming
4.6 Relative product and composition
Selected items from the Introduction to set theory and to me methodology and philosophy of mathematics and computer programing.
4.5 Inverse relation
This document introduces the concept of an inverse relation. It defines an inverse relation R-1 as the set containing all pairs (b,a) such that (a,b) is in R. It provides examples of inverse relations and shows how inverse relations can be defined using formulas, tuples, truth tables, and visual representations like graphs. The document also presents exercises for identifying inverse relations and propositions about inverse relations and their properties.
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6.1 Partitions
- 1. Introduction to set theory and to methodology and philosophy of mathematics and computer programming Partitions An overview by Jan Plaza c 2017 Jan Plaza Use under the Creative Commons Attribution 4.0 International License Version of December 6, 2017
- 2. Definition Let X be a family of sets. X is pairwise disjoint if every two different sets in X are disjoint. Definition Let X be a set. A partition of X is a family P of sets with the following properties. Cover: P = X, Disjointness: P is a pairwise disjoint family of sets, Non-empty components: ∅ ∈ P.
- 3. Example For i = 0, 1, 2, let Xi = {n : ∃k∈Z n = 3k + i}. The family {X0, X1, X2} is a partition of Z. Exercise 1. List all the partitions of {1, 2, 3}. 2. List all the partitions of {1, 2}. 3. List all the partitions of {1}. 4. List all the partitions of ∅.