Lecture 2.2: Set Theory*

1. Lecture 2.2: Set Theory* CS 250, Discrete Structures, Fall 2011 Nitesh Saxena *Adopted from previous lectures by Cinda Heeren

2. Course Admin • Slides from previous lectures all posted • HW1 Posted • Due at 11am 09/09/11 • Please follow all instructions • Recall: late submissions will not be accepted • Word Equation editor; Open Office; Alt-Codes • Please pick up your competency exams, if you haven’t done so Lecture 2.2 -- Set Theory

3. Outline • Set Theory, Operations and Laws Lecture 2.2 -- Set Theory

4. U A B like “exclusive or” 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) Lecture 2.2 -- Set Theory

5. 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) Lecture 2.2 -- Set Theory

6. Don’t memorize them, understand them! They’re in Rosen, p. 130 Set Theory - Famous Laws • Two pages of (almost) obvious. • One page of HS algebra. • One page of new. Lecture 2.2 -- Set Theory

7. A  U = A A U U = U A U A = A A U  = A A  =  A A = A Set Theory - Famous Laws • Identity • Domination • Idempotent Lecture 2.2 -- Set Theory

8. A U A = U A = A A A= Set Theory - Famous Laws • Excluded Middle • Uniqueness • Double complement Lecture 2.2 -- Set Theory

9. B U A B  A A U (B U C) A U (B C) = A  (B C) A  (B U C) = Set Theory – Famous Laws • Commutativity • Associativity • Distributivity A U B = A  B = (A U B)U C = (A  B) C = (A U B)  (A U C) (A  B) U (A  C) Lecture 2.2 -- Set Theory

10. (A UB)= A  B (A  B)= A U B Venn Diagrams are good for intuition, but we aim for a more formal proof. Set Theory – Famous Laws • DeMorgan’s I • DeMorgan’s II p q Lecture 2.2 -- Set Theory

11. New & important Like truth tables Not hard, a little tedious 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. Lecture 2.2 -- Set Theory

12. Example – the first way Prove that • () (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 UB)= A  B Lecture 2.2 -- Set Theory

13. (A UB)= A  B Example – the second way Prove that using a membership table. 0 : x is not in the specified set 1 : otherwise Lecture 2.2 -- Set Theory

14. (A UB)= A  B (A UB)= {x : (x  A v x  B)} = A  B = {x : (x  A)  (x  B)} Example – the third way Prove that using logically equivalent set definitions. = {x : (x  A)  (x  B)} Lecture 2.2 -- Set Theory

15. Another example: applying the laws X  (Y - Z) = (X  Y) - (X  Z). True or False? Prove your response. (X  Y) - (X  Z) = (X  Y)  (X  Z)’ = (X  Y)  (X’ U Z’) = (X  Y  X’) U (X  Y  Z’) =  U (X  Y  Z’) = (X  Y  Z’) Lecture 2.2 -- Set Theory

16. A U B =  A = B A  B =  A-B = B-A =  A Proof (direct and indirect) A  B =  Pv that if (A - B) U (B - A) = (A U B) then Suppose to the contrary, that A  B  , and that x  A  B. Then x cannot be in A-B and x cannot be in B-A. Then x is not in (A - B) U (B - A). But x is in A U B since (A  B)  (A U B). Thus, A  B = . Lecture 2.2 -- Set Theory

17. Today’s Reading • Rosen 2.1 and 2.2 Lecture 2.2 -- Set Theory