CS 173: Discrete Mathematical Structures

1. CS 173:Discrete Mathematical Structures Dan Cranston dcransto@uiuc.edu Rm 3240 Siebel Center Office Hours: Wed, Fri 11:30a-12:30p

2. CS 173 Announcements • Homework 2 returned this week (by email, in section, or in your section leader’s office hours). • Homework 3 available. Due 02/04, 8a.

3. Note:   {} Set Theory - Definitions and notation A set is an unordered collection of elements. Some examples: {1, 2, 3} is the set containing “1” and “2” and “3.” {1, 1, 2, 3, 3} = {1, 2, 3} since repetition is irrelevant. {1, 2, 3} = {3, 2, 1} since sets are unordered. {1, 2, 3, …} is a way we denote an infinite set (in this case, the natural numbers).  = {} is the empty set, or the set containing no elements.

4. A Venn Diagram B Set Theory - Definitions and notation x  S means “x is an element of set S.” x  S means “x is not an element of set S.” A  B means “A is a subset of B.” or, “B contains A.” or, “every element of A is also in B.” or, x ((x  A)  (x  B)).

5. Set Theory - Definitions and notation A  B means “A is a subset of B.” A  B means “A is a superset of B.” A = B if and only if A and B have exactly the same elements. iff, A  B and B  A iff, A  B and A  B iff, x ((x  A)  (x  B)). • So to show equality of sets A and B, show: • A  B • B  A

6. A B Set Theory - Definitions and notation A  B means “A is a proper subset of B.” • A  B, and A  B. • x ((x  A)  (x  B))  x ((x  B)  (x  A)) • x ((x  A)  (x  B)) x ((x  B) (x  A))

7. Vacuously Set Theory - Definitions and notation Quick examples: • {1,2,3}  {1,2,3,4,5} • {1,2,3}  {1,2,3,4,5} Is   {1,2,3}? Yes!x (x  )  (x  {1,2,3}) holds, because (x  ) is false. Is   {1,2,3}? No! Is   {,1,2,3}? Yes! Is   {,1,2,3}? Yes!

8. Yes Yes Yes No Set Theory - Definitions and notation Quiz time: Is {x}  {x}? Is {x}  {x,{x}}? Is {x}  {x,{x}}? Is {x}  {x}?

9. : and | are read “such that” or “where” Primes Set Theory - Ways to define sets • Explicitly: {John, Paul, George, Ringo} • Implicitly: {1,2,3,…}, or {2,3,5,7,11,13,17,…} • Set builder: { x : x is prime }, { x | x is odd }. In general { x : P(x) is true }, where P(x) is some description of the set. Ex. Let D(x,y) denote “x is divisible by y.” Give another name for { x : y ((y > 1)  (y < x)) D(x,y) }.

10. |S| = 3. |S| = 1. |S| = 0. |S| = 3. Set Theory - Cardinality If S is finite, then the cardinality of S, |S|, is the number of distinct elements in S. If S = {1,2,3}, If S = {3,3,3,3,3}, If S = , If S = { 1, {1}, {1,{1}} }, If S = {0,1,2,3,…}, |S| is infinity. (more on this later)

11. aka P(S) We say, “P(S) is the set of all subsets of S.” 2S = {, {a}}. 2S = {, {a}, {b}, {a,b}}. 2S = {}. 2S = {, {}, {{}}, {,{}}}. Set Theory - Power sets If S is a set, then the power set of S is 2S = { x : x  S }. If S = {a}, If S = {a,b}, If S = , If S = {,{}}, Fact: if S is finite, |P(S)| = 2|S|. (if |S| = n, |P(S)| = 2n)

12. A,B finite  |AxB| = ? • |A|+|B| • |A|-|B| • |A||B| Set Theory - Cartesian Product The Cartesian Product of two sets A and B is: A x B = { <a,b> : a  A  b  B} If A = {Charlie, Lucy, Linus}, and B = {Brown, VanPelt}, then A x B = {<Charlie, Brown>, <Lucy, Brown>, <Linus, Brown>, <Charlie, VanPelt>, <Lucy, VanPelt>, <Linus, VanPelt>} A1 x A2 x … x An = {<a1, a2,…, an>: a1  A1, a2  A2, …, an  An}

13. B A Set Theory - Operators The union of two sets A and B is: A  B = { x : x  A v x  B} If A = {Charlie, Lucy, Linus}, and B = {Lucy, Desi}, then A  B = {Charlie, Lucy, Linus, Desi}

14. B A Set Theory - Operators The intersection of two sets A and B is: A  B = { x : x  A  x  B} If A = {Charlie, Lucy, Linus}, and B = {Lucy, Desi}, then A  B = {Lucy}

15. B A Set Theory - Operators Another example If A = {x : x is a US president}, and B = {x : x is deceased}, then A  B = {x : x is a deceased US president}

16. B A Sets whose intersection is empty are called disjoint sets Set Theory - Operators One more example If A = {x : x is a US president}, and B = {x : x is in this room}, then A  B = {x : x is a US president in this room} = 

17. A • = U and U =  Set Theory - Operators The complement of a set A is: A = { x : x  A} If A = {x : x is bored}, then A = {x : x is not bored} U

18. U A B Set Theory - Operators The set difference, A - B, is: A - B = { x : x  A  x  B } A - B = A  B

19. 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)

20. 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)

21. Don’t memorize them, understand them! They’re in Rosen, p124. Set Theory - Famous Identities • Two pages of (almost) obvious. • One page of HS algebra. • One page of new.

22. A  U = A A U U = U A U A = A A U  = A A  =  A A = A Set Theory - Famous Identities • Identity • Domination • Idempotent

23. A U A = U A = A A A= Set Theory - Famous Identities • Excluded Middle • Uniqueness • Double complement

24. B U A B  A A U (B U C) A U (B C) = A  (B C) A  (B U C) = Set Theory - Famous Identities A U B = • Commutativity • Associativity • Distributivity A  B = (A U B)U C = (A  B) C = (A U B)  (A U C) (A  B) U (A  C)

25. (A UB)= A  B (A  B)= A U B Hand waving is good for intuition, but we aim for a more formal proof. Set Theory - Famous Identities • DeMorgan’s I • DeMorgan’s II p q

26. Like truth tables Like  Not hard, a little tedious Set Theory - 4 Ways to prove identities • Show that A  B and that A  B. • Use a membership table. • Use previously proven identities. • Use logical equivalences to prove equivalent set definitions.

27. (A UB)= A  B Set Theory - 4 Ways to prove identities 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)

28. (A B)= A U B Set Theory - 4 Ways to prove identities Prove that

29. (A UB)= A  B Haven’t we seen this before? Set Theory - 4 Ways to prove identities Prove that using a membership table. 0 : x is not in the specified set 1 : otherwise

30. (A UB)= A  B (A UB)= {x : (x  A v x  B)} = A  B = {x : (x  A)  (x  B)} Set Theory - 4 Ways to prove identities Prove that using logically equivalent set definitions. = {x : (x  A)  (x  B)}

31. Set Theory - A proof for you to do 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’)

32. A U B =  A = B A  B =  A-B = B-A =  Trying to pv p --> q Assume p and not q, and find a contradiction. Our contradiction was that sets weren’t equal. Set Theory - A proof for us to do together. A  B =  Prove 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. So 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 = .