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Set Theory BY Miss Manita Modphai

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©1999 Indiana University Trustees Why Study Set Theory? Understanding set theory helps people to … see things in terms of systems organize things into groups begin to understand logic

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©1999 Indiana University Trustees Key Mathematicians These mathematicians influenced the development of set theory and logic: Georg Cantor John Venn George Boole Augustus DeMorgan

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©1999 Indiana University Trustees Georg Cantor 1845 -1918 developed set theory set theory was not initially accepted because it was radically different set theory today is widely accepted and is used in many areas of mathematics

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©1999 Indiana University Trustees …Cantor the concept of infinity was expanded by Cantor’s set theory Cantor proved there are “levels of infinity” an infinitude of integers initially ending with or an infinitude of real numbers exist between 1 and 2; there are more real numbers than there are integers…

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©1999 Indiana University Trustees John Venn 1834-1923 studied and taught logic and probability theory articulated Boole’s algebra of logic devised a simple way to diagram set operations (Venn Diagrams)

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©1999 Indiana University Trustees George Boole 1815-1864 self ‑ taught mathematician with an interest in logic developed an algebra of logic (Boolean Algebra) featured the operators –and –or –not –nor (exclusive or)

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©1999 Indiana University Trustees Augustus De Morgan 1806-1871 developed two laws of negation interested, like other mathematicians, in using mathematics to demonstrate logic furthered Boole’s work of incorporating logic and mathematics formally stated the laws of set theory

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©1999 Indiana University Trustees Basic Set Theory Definitions A set is a collection of elements An element is an object contained in a set If every element of Set A is also contained in Set B, then Set A is a subset of Set B –A is a proper subset of B if B has more elements than A does The universal set contains all of the elements relevant to a given discussion

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©1999 Indiana University Trustees Simple Set Example the universal set is a deck of ordinary playing cards each card is an element in the universal set some subsets are: –face cards –numbered cards –suits –poker hands

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©1999 Indiana University Trustees Set Theory Notation SymbolMeaning Upper casedesignates set name Lower casedesignates set elements { }enclose elements in set or is (or is not) an element of is a subset of (includes equal sets) is a proper subset of is not a subset of is a superset of | or :such that (if a condition is true) | |the cardinality of a set

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©1999 Indiana University Trustees Set Notation: Defining Sets a set is a collection of objects sets can be defined two ways: –by listing each element –by defining the rules for membership Examples: –A = {2,4,6,8,10} –A = {x|x is a positive even integer <12}

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©1999 Indiana University Trustees Set Notation Elements an element is a member of a set notation: means “is an element of” means “is not an element of” Examples: –A = {1, 2, 3, 4} 1 A6 A 2 Az A –B = {x | x is an even number 10} 2 B9 B 4 Bz B

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©1999 Indiana University Trustees Subsets a subset exists when a set’s members are also contained in another set notation: means “is a subset of” means “is a proper subset of” means “is not a subset of”

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©1999 Indiana University Trustees Subset Relationships A = {x | x is a positive integer 8} set A contains: 1, 2, 3, 4, 5, 6, 7, 8 B = {x | x is a positive even integer 10} set B contains: 2, 4, 6, 8 C = {2, 4, 6, 8, 10} set C contains: 2, 4, 6, 8, 10 Subset Relationships A AA BA C B AB BB C C AC BC C

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©1999 Indiana University Trustees Set Equality Two sets are equal if and only if they contain precisely the same elements. The order in which the elements are listed is unimportant. Elements may be repeated in set definitions without increasing the size of the sets. Examples: A = {1, 2, 3, 4} B = {1, 4, 2, 3} A B and B A; therefore, A = B and B = A A = {1, 2, 2, 3, 4, 1, 2} B = {1, 2, 3, 4} A B and B A; therefore, A = B and B = A

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©1999 Indiana University Trustees Cardinality of Sets Cardinality refers to the number of elements in a set A finite set has a countable number of elements An infinite set has at least as many elements as the set of natural numbers notation: |A| represents the cardinality of Set A

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©1999 Indiana University Trustees Finite Set Cardinality Set Definition Cardinality A = {x | x is a lower case letter} |A| = 26 B = {2, 3, 4, 5, 6, 7} |B| = 6 C = {x | x is an even number 10} |C|= 4 D = {x | x is an even number 10} |D| = 5

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©1999 Indiana University Trustees Infinite Set Cardinality Set DefinitionCardinality A = {1, 2, 3, …}|A| = B = {x | x is a point on a line}|B| = C = {x| x is a point in a plane} |C| =

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©1999 Indiana University Trustees Universal Sets The universal set is the set of all things pertinent to to a given discussion and is designated by the symbol U Example: U = {all students at IUPUI} Some Subsets: A = {all Computer Technology students} B = {freshmen students} C = {sophomore students}

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©1999 Indiana University Trustees The Empty Set Any set that contains no elements is called the empty set the empty set is a subset of every set including itself notation: { } or Examples ~ both A and B are empty A = {x | x is a Chevrolet Mustang} B = {x | x is a positive number 0}

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©1999 Indiana University Trustees The Power Set ( P ) The power set is the set of all subsets that can be created from a given set The cardinality of the power set is 2 to the power of the given set’s cardinality notation: P ( set name) Example: A = {a, b, c}where |A| = 3 P (A) = {{a, b}, {a, c}, {b, c}, {a}, {b}, {c}, A, } and | P (A)| = 8 In general, if |A| = n, then | P (A) | = 2 n

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©1999 Indiana University Trustees Special Sets Z represents the set of integers –Z + is the set of positive integers and –Z - is the set of negative integers N represents the set of natural numbers n ℝ represents the set of real numbers Q represents the set of rational numbers

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©1999 Indiana University Trustees Venn Diagrams Venn diagrams show relationships between sets and their elements Universal Set Sets A & B

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©1999 Indiana University Trustees Venn Diagram Example 1 Set DefinitionElements A = {x | x Z + and x 8}1 2 3 4 5 6 7 8 B = {x | x Z + ; x is even and 10}2 4 6 8 10 A B B A

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©1999 Indiana University Trustees Venn Diagram Example 2 Set DefinitionElements A = {x | x Z + and x 9} 1 2 3 4 5 6 7 8 9 B = {x | x Z + ; x is even and 8}2 4 6 8 A B B A A B

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©1999 Indiana University Trustees Venn Diagram Example 3 Set DefinitionElements A = {x | x Z + ; x is even and 10} 2 4 6 8 10 B = x Z + ; x is odd and x 10 } 1 3 5 7 9 A B B A

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©1999 Indiana University Trustees Venn Diagram Example 4 Set Definition U = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 2, 6, 7} B = {2, 3, 4, 7} C = {4, 5, 6, 7} A = {1, 2, 6, 7}

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©1999 Indiana University Trustees Venn Diagram Example 5 Set Definition U = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 2, 6, 7} B = {2, 3, 4, 7} C = {4, 5, 6, 7} B = {2, 3, 4, 7}

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©1999 Indiana University Trustees Venn Diagram Example 6 Set Definition U = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 2, 6, 7} B = {2, 3, 4, 7} C = {4, 5, 6, 7}

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