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Chapter 6: Set Theory 6.1 Set Theory: Definitions and the Element Method of Proof Set Theory - Definitions and the Element Method of Proof The introduction of suitable abstractions is our only mental aid to organize and master complexity. – E. W. Dijkstra, 1930 – 2002 Erickson

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Let’s write what it means for a set A to be a subset of a set B as a formal universal conditional statement: 6.1 Set Theory - Definitions and the Element Method of Proof 2 A B x, if x A then x B. Erickson

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The negation is existential 6.1 Set Theory - Definitions and the Element Method of Proof 3 A B x, if x A and x B. Erickson

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A proper subset of a set is a subset that is not equal to its containing set. 6.1 Set Theory - Definitions and the Element Method of Proof 4 A is a proper subset of B 1. A B, and 2. there is at least one element in B that is not in A. Erickson

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Let sets X and Y be given. To prove that X Y, 1.Suppose that x is a particular but arbitrarily chosen element of X, 2.Show that x is an element of Y. 6.1 Set Theory - Definitions and the Element Method of Proof 5Erickson

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Let A = {n | n = 5r for some integer r} and B = {m | m = 20s for some integer s}. a.Is A B? Explain. b.Is B A? Explain. 6.1 Set Theory - Definitions and the Element Method of Proof 6Erickson

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Given sets A and B, A equals B, written A = B, iff every element of A is in B and every element of B is in A. Symbolically, 6.1 Set Theory - Definitions and the Element Method of Proof 7 A = B A B and B A Erickson

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Let A and B be subsets of a universal set U. 6.1 Set Theory - Definitions and the Element Method of Proof 8 1. The union of A and B denoted A B, is the set of all elements that are in at least one of A or B. Symbolically: A B = {x U | x A or x B} Erickson

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Let A and B be subsets of a universal set U. 6.1 Set Theory - Definitions and the Element Method of Proof 9 2. The intersection of A and B denoted A B, is the set of all elements that are common to both A or B. Symbolically: A B = {x U | x A and x B} Erickson

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Let A and B be subsets of a universal set U. 6.1 Set Theory - Definitions and the Element Method of Proof The difference of B minus A (or relative complement of A in B) denoted B – A, is the set of all elements that are in B but not A. Symbolically: B – A = {x U | x B and x A} Erickson

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Let A and B be subsets of a universal set U. 6.1 Set Theory - Definitions and the Element Method of Proof The complement of A denoted A c, is the set of all elements in U that are not A. Symbolically: A c = {x U | x A} Erickson

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Let the universal set be the set R of all real numbers and let A = {x R | 0 < x 2}, B = {x R | 1 x < 4}, and C = {x R | 3 x < 9}. Find each of the following: a. A Bb. A Bc. A c d. A C e. A Cf. B c g. A c B c h. A c B c i. (A B) c j. (A B) c 6.1 Set Theory - Definitions and the Element Method of Proof 12Erickson

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Given sets A 0, A 1, A 2, … that are subsets of a universal set U and given a nonnegative integer n, 6.1 Set Theory - Definitions and the Element Method of Proof 13Erickson

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Empty Set A set with no elements is called the empty set (or null set) and denoted by the symbol . Disjoint Two sets are called disjoint iff they have no elements in common. Symbolically: A and B are disjoint A B = 6.1 Set Theory - Definitions and the Element Method of Proof 14Erickson

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Mutually Disjoint Sets A 1, A 2, A 3, … are mutually disjoint (or pairwise disjoint or nonoverlapping) iff no two sets A i and A j with distinct subscripts have any elements in common. More precisely, for all i, j = 1, 2, 3, … A i A j = whenever i j. 6.1 Set Theory - Definitions and the Element Method of Proof 15Erickson

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Let for all positive integers i. 6.1 Set Theory - Definitions and the Element Method of Proof 16Erickson

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Partition A finite or infinite collection of nonempty sets {A 1, A 2, A 3, …} is a partition of a set A iff, 1. A is the union of all the A i 2.The sets A 1, A 2, A 3, …are mutually disjoint. 6.1 Set Theory - Definitions and the Element Method of Proof 17Erickson

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6.1 Set Theory - Definitions and the Element Method of Proof 18Erickson

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Power Set Given a set A, the power set of A is denoted (A), is the set of all subsets of A. 6.1 Set Theory - Definitions and the Element Method of Proof 19Erickson

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Suppose A = {1, 2} and B = {2, 3}. Find each of the following: 6.1 Set Theory - Definitions and the Element Method of Proof 20Erickson

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