Lecture 2.2: Set Theory CS 250, Discrete Structures, Fall 2015 Nitesh Saxena Adopted from previous lectures by Cinda Heeren

1/6/2016Lecture Set Theory2 Course Admin HW1 Was just due We will start to grade it We will provide a solution set soon Word Equation editor; Open Office Travel next week Attending and presenting at a conference in Vienna: No class next week (Tuesday and Thursday) Would not affect our coverage Please utilize this time to review the previous lectures

1/6/2016Lecture Set Theory3 Outline Set Theory, Operations and Laws

1/6/2016Lecture Set Theory4 Set Theory - Operators The symmetric difference, A  B, is: A  B = { x : (x  A  x  B) v (x  B  x  A)} = (A - B) U (B - A) like “exclusive or” A U B

1/6/2016Lecture Set Theory5 Set Theory - Operators A  B = { x : (x  A  x  B) v (x  B  x  A)} = (A - B) U (B - A) Proof:{ x : (x  A  x  B) v (x  B  x  A)} = { x : (x  A - B) v (x  B - A)} = { x : x  ((A - B) U (B - A))} = (A - B) U (B - A)

1/6/2016Lecture Set Theory6 Set Theory - Famous Laws Two pages of (almost) obvious. One page of HS algebra. One page of new. Don’t memorize them, understand them! They’re in Rosen, p. 130

1/6/2016Lecture Set Theory7 Set Theory - Famous Laws Identity Domination Idempotent A  U = A A U  = A A U U = U A   =  A U A = A A  A = A

1/6/2016Lecture Set Theory8 Set Theory - Famous Laws Excluded Middle Uniqueness Double complement A U A = U A  A =  A = A

1/6/2016Lecture Set Theory9 Set Theory – Famous Laws Commutativity Associativity Distributivity A U B = (A U B) U C = A  B = B U A B  A (A  B)  C = A U (B U C) A  (B  C) A U (B  C) = A  (B U C) = (A U B)  (A U C) (A  B) U (A  C)

1/6/2016Lecture Set Theory10 Set Theory – Famous Laws DeMorgan’s I DeMorgan’s II pq Venn Diagrams are good for intuition, but we aim for a more formal proof. (A U B) = A  B (A  B) = A U B

1/6/2016Lecture Set Theory11 3 Ways to prove Laws or set equalities Show that A  B and that A  B. Use a membership table. Use logical equivalences to prove equivalent set definitions. New & importantLike truth tablesNot hard, a little tedious

1/6/2016Lecture Set Theory12 Example – the first way Prove that 1. (  ) (x  A U B)  (x  A U B)  (x  A and x  B)  (x  A  B) 2. (  ) (x  A  B)  (x  A and x  B)  (x  A U B)  (x  A U B) (A U B) = A  B

1/6/2016Lecture Set Theory13 Example – the second way Prove that using a membership table. 0 : x is not in the specified set 1 : otherwise (A U B) = A  B ABABA  BAUB

1/6/2016Lecture Set Theory14 Example – the third way Prove that using logically equivalent set definitions. (A U B) = A  B (A U B) = {x :  (x  A v x  B)} = {x :  (x  A)   (x  B)} = A  B = {x : (x  A)  (x  B)}

1/6/2016Lecture Set Theory15 Another example: applying the laws X  (Y - Z) = (X  Y) - (X  Z). True or False? Prove your response. = (X  Y)  (X’ U Z’) = (X  Y  X’) U (X  Y  Z’) =  U (X  Y  Z’) = (X  Y  Z’) = X  (Y - Z) (X  Y) - (X  Z) = (X  Y)  (X  Z)’

1/6/2016Lecture Set Theory16 Suppose to the contrary, that A  B  , and that x  A  B. A Proof (direct and indirect) Pv that if (A - B) U (B - A) = (A U B) then Then x cannot be in A-B and x cannot be in B-A. But x is in A U B since (A  B)  (A U B). A  B =  Thus, A  B = . Then x is not in (A - B) U (B - A).

1/6/2016Lecture Set Theory17 Today’s Reading Rosen 2.1 and 2.2