Sets
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This document discusses sets and their properties. It defines what a set is and the different ways to describe a set using word descriptions, listings, or set-builder notation. It discusses elements and members of sets, and whether certain values are elements of example sets. It also defines key set concepts like cardinality (the number of elements in a set), finite vs infinite sets, and why cardinal numbers cannot be determined for infinite sets.
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This document covers key concepts about sets including:
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The document defines and explains various concepts related to sets. It discusses how sets were developed by Georg Cantor and defines elements and representation of sets using roster and set builder forms. It also defines and provides examples of empty, finite, infinite, equal, subset, power and universal sets. Finally, it explains operations of sets such as union, intersection and complements and provides examples of each.
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This document discusses key concepts in set theory including:
- Sets are collections of objects that can be represented by listing elements or using set-builder notation.
- A set is well-defined if its elements can clearly be identified. Sets can be finite or infinite depending on the number of elements.
- Subsets, unions, intersections, complements and other relationships between sets are defined using set notation and set-builder notation.
- Examples are provided to demonstrate concepts like subsets, unions, intersections, and well-defined sets.
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1) Set theory helps organize things into groups and understand logic. Key contributors include Georg Cantor, John Venn, George Boole, and Augustus DeMorgan.
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3) Venn diagrams show relationships between sets using overlapping circles to represent their common elements.
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Sets and their representations
Two methods of representing a set:
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Finite and infinite sets
Consider some examples
Equal sets
Subsets
Subsets of set of real numbers:
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Universal Set
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- A set is well-defined if its elements can clearly be identified. Sets can be finite or infinite depending on the number of elements.
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Sets and their representations
Two methods of representing a set:
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The empty set
Finite and infinite sets
Consider some examples
Equal sets
Subsets
Subsets of set of real numbers:
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Universal Set
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The answer for:
1)Give me a group of girls whose height is > than 156 cm is E,F,G.
2) The answers for Piano and Guitar question is:
n(U) =8,
n(A)=3,
n(B)=4
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The document defines set concepts and their properties. It explains that a set is a collection of distinct objects, called elements. Properties discussed include how sets are denoted, elements belong or don't belong to sets, order doesn't matter, and counting each element once. It also covers set theory, Venn diagrams, ways to represent sets, types of sets (empty, finite, infinite), equal sets, subsets, cardinality, and operations like intersection, union, difference, and complement of sets. Examples are provided to illustrate each concept.
7-Sets-1.ppt
This document provides an overview of sets and related concepts in discrete mathematics. Some key points covered include:
- A set is an unordered collection of distinct objects. Sets can contain numbers, words, or other sets. Order and duplicates do not matter.
- Sets are specified using curly brackets and listing elements, set-builder notation, ellipses, or capital letters. Membership is denoted using the symbol ∈.
- Basic set relationships include subsets, proper subsets, equality, the empty set, unions, and intersections. Power sets contain all possible subsets.
- Tuples are ordered lists used to specify locations in n-dimensional spaces. Cartesian products combine elements from multiple sets into ordered pairs
02 Representing Sets and Types of Sets.pptx
This document provides information about visually representing sets using different methods. It discusses listing sets using curly brackets, representing sets using set-builder notation with conditions, and displaying sets using Venn diagrams. The document also covers labeling sets, finding cardinality, types of sets such as empty, finite, infinite, and singleton sets, and relationships between sets like equal, equivalent, subset, and superset.
Explore the foundational concepts of sets in discrete mathematics
Explore the foundational concepts of sets in discrete mathematics with this comprehensive PowerPoint presentation. Whether you are a student delving into the world of discrete structures or an enthusiast eager to understand the fundamentals, this presentation serves as an insightful guide.
Shri Ramswaroop Memorial University.pptx
This document discusses sets and set theory. It defines what a set is, the different types of sets, and how sets are represented. The key points are:
- A set is a collection of distinct objects, called elements or members. Sets can be finite or infinite.
- There are different types of sets including empty sets, which have no elements, singleton sets, which have one element, and finite sets, which have a fixed number of elements.
- Sets can be represented in roster form by listing elements between curly brackets or in set builder form using a rule to describe the elements.
- Equivalent sets have the same number of elements, while infinite sets have an indefinite number of
INTRODUCTION TO SETS.pptx
The document introduces basic concepts of set theory, including:
- A set is a collection of distinct objects called elements or members.
- Special sets include the natural numbers, integers, rational numbers, and real numbers.
- Types of sets include subsets, equal sets, empty sets, singleton sets, finite sets, infinite sets, disjoint sets, power sets, and universal sets.
- Cardinal numbers represent the number of elements in a set.
#17 Boba PPT Template (1).pptx
This document defines sets and their properties. It begins by defining a set as a well-defined collection of objects and notes that sets are conventionally denoted with capital letters while elements are denoted with lowercase letters. It then discusses different ways of representing sets, including statement form, roster form, and set builder form. Venn diagrams are introduced as a pictorial way to represent sets and relationships between sets. Various set symbols and their meanings are defined. Different types of sets such as singleton, finite, and infinite sets are described. Properties of sets like commutative, associative, distributive, identity, complement, idempotent, bound, and absorption laws are explained. Finally, operations on sets like union, intersection, difference,
Sets.pdf
This document defines and provides examples of sets. It explains that a set is a collection of distinct objects called elements or members. Sets can be represented symbolically using uppercase letters for the set and lowercase letters for elements. Common set operations like union and intersection are introduced along with properties like commutativity, associativity, and idempotence. Venn diagrams are described as a way to visually represent relationships between sets. Examples of different types of number sets such as natural, integer, and rational numbers are also given.
Moazzzim Sir (25.07.23)CSE 1201, Week#3, Lecture#7.pptx
This document provides an overview of key concepts in set theory including:
- The definition of a set as an unordered collection of distinct elements
- Common ways to describe and represent sets such as listing elements, set-builder notation, and Venn diagrams
- Important set terminology including subset, proper subset, set equality, cardinality (size of a set), finite vs infinite sets, power set, and Cartesian product
The document uses examples and explanations to illustrate each concept over 34 pages. It appears to be lecture material introducing students to the basic foundations of set theory.
Chapter 2 Mathematical Language and Symbols.pdf
This document discusses mathematical language and symbols. It defines key concepts such as sets, relations, functions, and binary operations. Sets are collections of distinct objects that can be defined using a roster or rule. Relations pair elements between two sets. A function is a special type of relation where each input is paired with exactly one output. Binary operations take two inputs from a set and return an output in that same set. Common properties of binary operations include commutativity and associativity.
Lesson2_MathematicalLanguageAndSymbols _Lesson 2.1_VariablesAndTheLanguageOfS...
Lesson2_MathematicalLanguageAndSymbols _Lesson 2.1_VariablesAndTheLanguageOfSets.pdf
This pdf tackles about the Mathematical Language and Symbols and the Variables and the Language of Sets.
This presentation contains definitions, tables, illustrations as well as examples.
I hope you'll find this helpful.
Sets of numbers
Here are the translations between set notations:
A. The set of integers greater than -5.5
B. {10, 20, 30, 40, ...}
C. {x | x ≤ -2}
Mathematics 7 week 1
There are 4 groups. Each group belongs to a group. The set is an object that belongs to more than one group. The document discusses sets, their definitions, notations, types (finite, infinite, empty set), cardinality, and equivalence versus equality of sets.
Discrete Structure Mathematics lecture 1
The document summarizes key concepts in discrete mathematics including sets, operations on sets, functions, sequences, and counting techniques. It defines what a set is, ways to describe sets, and set operations like unions and intersections. Examples are given of common sets like integers, rational numbers, and real numbers. Subsets, the empty set, cardinality (size) of sets, and Venn diagrams are also explained.
Discrete Math Chapter 2: Basic Structures: Sets, Functions, Sequences, Sums, ...
1) The document summarizes key concepts from a discrete mathematics course including sets, operations on sets, functions, sequences, and sums.
2) It covers topics like basic set theory, operations on sets like union and intersection, subsets, power sets, Cartesian products, and cardinality.
3) Examples are provided to illustrate concepts like Venn diagrams, disjoint sets, complements, and set differences.
Sets and Functions By Saleh ElShehabey
The document discusses sets and set notation. It begins by defining a set as a collection of elements and noting that sets are fundamental data structures in mathematics. It then provides examples of set notation using curly braces and discusses properties of sets such as that a set is defined by its elements regardless of ordering or repetition. The document also introduces concepts like subsets, membership using the Greek letter epsilon, the empty set, cardinality, and power sets.
Set theory- Introduction, symbols with its meaning
The document provides information about the concepts of set theory that will be covered in the Discrete Mathematics and Information Theory course. It defines basic set operations like union, intersection, complement and subset. It explains notation for set membership, empty set, universal set and Venn diagrams. Examples are given for each concept to illustrate the set operations and relationships between different sets.
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Explore the foundational concepts of sets in discrete mathematics
Explore the foundational concepts of sets in discrete mathematics
Moazzzim Sir (25.07.23)CSE 1201, Week#3, Lecture#7.pptx
Moazzzim Sir (25.07.23)CSE 1201, Week#3, Lecture#7.pptx
Lesson2_MathematicalLanguageAndSymbols _Lesson 2.1_VariablesAndTheLanguageOfS...
Lesson2_MathematicalLanguageAndSymbols _Lesson 2.1_VariablesAndTheLanguageOfS...
Discrete Math Chapter 2: Basic Structures: Sets, Functions, Sequences, Sums, ...
Discrete Math Chapter 2: Basic Structures: Sets, Functions, Sequences, Sums, ...
Set theory- Introduction, symbols with its meaning
Set theory- Introduction, symbols with its meaning
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Infinite sets and cardinalities
The document discusses different types of infinite sets and their cardinal numbers. It provides examples of sets that are countable, like the natural numbers and integers, by showing a one-to-one correspondence between each set and a subset. However, the set of real numbers is uncountable because there is an infinite number of decimals between any two real numbers, making it impossible to list them in a one-to-one fashion. Infinite sets can be defined as those that can be placed in a one-to-one correspondence with a proper subset of themselves.
Intro to logic
This document introduces some key concepts in symbolic logic including:
1. Statements are declarative sentences that can be either true or false, while non-examples include opinions, questions, paradoxes, and commands.
2. Compound statements are formed using logical connectives like AND, OR, and IF...THEN to join two or more statements.
3. Negations are formed by introducing the word "not" to a statement, and the negation of a true statement is false while the negation of a false statement is true.
4. Quantifiers like "all", "some", and "no" are used in mathematics to indicate how many of a particular situation exist.
Operations with sets
The document discusses set operations including unions, intersections, differences of sets, and Cartesian products. Ordered pairs are used in Cartesian products where order matters, unlike sets where order does not. Cartesian products pair each element of one set with each element of another set. The cardinal number, or total number of elements, of a Cartesian product of two sets A and B is found by multiplying the cardinal numbers of each set, written as n(A) x n(B). Venn diagrams can be used to represent set operations visually.
Solving problems by inductive reasoning
This document discusses making logical arguments and general conclusions from observations. It explains that while repeated observations of specific examples can lead to general principles, these principles do not always apply to all cases. Applying general conclusions as rules to new specific examples may not always yield accurate predictions, as the available data is limited and exceptions may exist.
Problem solving strategies
The document provides tips and strategies for solving problem-solving problems, including working backwards, trial and error, guessing and checking, looking for patterns in similar problems, drawing pictures, avoiding getting stuck on surface-level details, and using common sense. It discusses the Fibonacci sequence, provides an example of working backwards on a racetrack problem, and advises drawing pictures to organize thoughts when representing physical objects.
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This document discusses estimation and story problems. It provides examples of estimating the number of busses needed for a field trip with 115 people where each bus holds 27 people. It recommends always rounding up when estimating. It also gives an example of estimating a baseball player's batting average by rounding his at bats to 600 and hits to 200 to get an estimated average of 0.333. Finally, it mentions using proportions to compare two different ratios.
Sets copy
This document discusses sets and their properties. It defines what a set is and the different ways to describe a set using word descriptions, listings, or set-builder notation. It discusses elements and members of sets, and whether certain values are elements of example sets. It also defines key set concepts like cardinality (the number of elements in a set), finite vs infinite sets, and why cardinal numbers cannot be determined for infinite sets.
Using inductive reasoning
This document discusses inductive reasoning and using number patterns to derive mathematical rules. It introduces key concepts like number sequences, terms, arithmetic sequences with a common difference, and geometric sequences with a common ratio. It explains using a tree diagram of successive differences to identify patterns in number sequences and deduce rules. As an example, it shows deriving the formula for finding the sum of the first n odd counting numbers as n^2.
Sets, subsets, compliments
The document defines various types of numbers and sets them in a circle chart. It then discusses concepts related to sets such as complements, proper subsets, and the formula to calculate the number of subsets in a set based on the number of elements. The key points are:
1) Natural numbers are contained within whole numbers, integers, rational numbers, real numbers, and imaginary numbers in descending order.
2) The complement of a set A is everything in the universal set that is not in A.
3) A proper subset has some but not all elements of the original set.
4) The number of subsets of a set is calculated by 2 to the power of the number of elements (2
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Sets
- 1. Set Theory Sets, elements, kinds of numbers