SET

  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.

  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}

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.

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.

Venn diagrams ΩΩΩΩ B B A

Set Operations   Complementation   A c = {  :   A}   Union   A ∪ B = {  :  A or  B (or both)}   Intersection   A ∩ B = {  :  A and  B}

  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

  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

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

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