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SET

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A set is a collection of elements. Sets are usually denoted by capital letters A, B, Ω, etc. Elements are usually denoted by lower case letters x, y, ω, etc. If x is an element of a set A, we write x A. If x is an not element of a set A, we write x A.

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A set can be described by listing the set’s elements A = {1, 2, 3, 4} B = {apple, pear, orange} by describing the set in words “A is the set of all real numbers between 0 and 1, inclusive.” by using the notation {ω: specification for ω} A = {x : 0 x 1} or sometimes we simply write A = {0 x 1}

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Two important sets The universe is the set containing all points under consideration and is denoted by Ω. The empty set (the set containing no elements) is denoted by Ø.

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Set relations If every point of set A belongs to set B, then we say that A is a subset of B (B is a superset of A). We write A B B A A = B if and only if A B and B A.

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Venn diagrams ΩΩΩΩ B B A

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Set Operations Complementation A c = { : A} Union A ∪ B = { : A or B (or both)} Intersection A ∩ B = { : A and B}

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Two sets A and B are disjoint (or mutually exclusive) if A ∩ B = Ø. These basic operations can be extended to any finite number of sets. A ∪ B ∪ C = A ∪ (B ∪ C) = (A ∪ B) ∪ C and A ∩ B ∩C = A ∩ (B ∩ C) = (A ∩ B) ∩C

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You can show that (a) A ∪ B = B ∪ A (b) A ∩ B = B ∩ A (c) A ∪ (B ∪ C) = (A ∪ B) ∪ C (d) A ∩ (B ∩ C) = (A ∩ B) ∩ C Note: (a) and (b) are the commutative laws Note: (c) and (d) are the associative laws

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More Set Identities (e) A ∪ (B∩ C) = (A ∪ B) ∩ (A ∪ C) (f) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (g) A ∩ Ø = Ø (h) A ∪ Ø = A (i) (A ∪ B) c = A c ∩ B c (j) (A ∩ B) c = A c ∪ B c (k) (A c ) c = A Note: (i) and (j) are called DeMorgan’s Laws

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Proving statements about sets If A B and B C, then A C. Proof: Discussion Reasons (1) If x A then x B Def. of A B (2) Since x B then x C Def. of B C (3) If x A then x C By stmts (1) and (2) (4) Therefore A C Def. of A C

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