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CS 173: Discrete Mathematical Structures

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  1. CS 173:Discrete Mathematical Structures Cinda Heeren heeren@cs.uiuc.edu Rm 2213 Siebel Center Office Hours: W 9:30-11:30a

  2. For breakfast I ate: • cereal • eggs • brew • donuts CS 173 Announcements • Homework 2… • Homework 3 available. Due 02/05, 8a. • Midterm 1: 2/23/06, 7-9p, location TBA. • Let’s try the remotes… Cs173 - Spring 2004

  3. Note:   {} CS 173 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. Cs173 - Spring 2004

  4. A Venn Diagram B CS 173 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)). Cs173 - Spring 2004

  5. CS 173 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 Cs173 - Spring 2004

  6. A B CS 173 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) v (x  A)) • x ((x  A)  (x  B)) x ((x  B) (x  A)) • x ((x  A)  (x  B)) x ((x  B)  (x  A)) Cs173 - Spring 2004

  7. Vacuously CS 173 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! Cs173 - Spring 2004

  8. Yes Yes Yes No CS 173 Set Theory - Definitions and notation Quiz time: Is {x}  {x}? Is {x}  {x,{x}}? Is {x}  {x,{x}}? Is {x}  {x}? Cs173 - Spring 2004

  9. : and | are read “such that” or “where” Primes CS 173 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) }. Can we use any predicate P to define a set S = { x : P(x) }? Cs173 - Spring 2004

  10. So S must not be in S, right? ARRRGH! No! Who shaves the barber? CS 173 Set Theory - Russell’s Paradox Can we use any predicate P to define a set S = { x : P(x) }? Define S = { x : x is a set where x  x } Then, if S  S, then by defn of S, S  S. But, if S  S, then by defn of S, S  S. There is a town with a barber who shaves all the people (and only the people) who don’t shave themselves. Cs173 - Spring 2004

  11. |S| = 3. |S| = 1. |S| = 0. |S| = 3. CS 173 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 = { , {}, {,{}} }, If S = {0,1,2,3,…}, |S| is infinite. (more on this later) Cs173 - Spring 2004

  12. aka P(S) We say, “P(S) is the set of all subsets of S.” 2S = {, {a}}. 2S = {, {a}, {b}, {a,b}}. 2S = {}. 2S = {, {}, {{}}, {,{}}}. CS 173 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, |2S| = 2|S|. (if |S| = n, |2S| = 2n) Cs173 - Spring 2004

  13. A,B finite  |AxB| = ? We’ll use these special sets soon! AxB |A|+|B| |A+B| |A||B| CS 173 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} Cs173 - Spring 2004

  14. B A CS 173 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} Cs173 - Spring 2004

  15. B A CS 173 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} Cs173 - Spring 2004

  16. CS 173 Set Theory - Operators The intersection of two sets A and B is: A  B = { x : x  A  x  B} If A = {x : x is a US president}, and B = {x : x is deceased}, then A  B = {x : x is a deceased US president} B A Cs173 - Spring 2004

  17. B A Sets whose intersection is empty are called disjoint sets CS 173 Set Theory - Operators The intersection of two sets A and B is: A  B = { x : x  A  x  B} 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} =  Cs173 - Spring 2004

  18. A • = U and U =  CS 173 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 Cs173 - Spring 2004

  19. U A B CS 173 Set Theory - Operators The set difference, A - B, is: A - B = { x : x  A  x  B } A - B = A  B Cs173 - Spring 2004

  20. U A B like “exclusive or” CS 173 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) Cs173 - Spring 2004

  21. CS 173 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) Cs173 - Spring 2004

  22. Don’t memorize them, understand them! They’re in Rosen, p89. CS 173 Set Theory - Famous Identities • Two pages of (almost) obvious. • One page of HS algebra. • One page of new. Cs173 - Spring 2004

  23. A  U = A A U U = U A U A = A A U  = A A  = A A A = A (Lazy) CS 173 Set Theory - Famous Identities • Identity • Domination • Idempotent Cs173 - Spring 2004

  24. A U A = U A = A A A= CS 173 Set Theory - Famous Identities • Excluded Middle • Uniqueness • Double complement Cs173 - Spring 2004

  25. B U A B  A A U (B U C) A U (B C) = A  (B C) A  (B U C) = CS 173 Set Theory - Famous Identities • 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) Cs173 - Spring 2004

  26. (A UB)= A  B (A  B)= A U B Hand waving is good for intuition, but we aim for a more formal proof. CS 173 Set Theory - Famous Identities • DeMorgan’s I • DeMorgan’s II p q Cs173 - Spring 2004

  27. New & important Like truth tables Like  Not hard, a little tedious CS 173 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. Cs173 - Spring 2004

  28. (A UB)= A  B CS 173 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) Cs173 - Spring 2004

  29. (A UB)= A  B Haven’t we seen this before? CS 173 Set Theory - 4 Ways to prove identities Prove that using a membership table. 0 : x is not in the specified set 1 : otherwise Cs173 - Spring 2004

  30. (A UB)= A  B (A UB)= A U B = A  B = A  B CS 173 Set Theory - 4 Ways to prove identities Prove that using identities. Cs173 - Spring 2004

  31. (A UB)= A  B (A UB)= {x : (x  A v x  B)} = A  B = {x : (x  A)  (x  B)} CS 173 Set Theory - 4 Ways to prove identities Prove that using logically equivalent set definitions. = {x : (x  A)  (x  B)} Cs173 - Spring 2004

  32. CS 173 Set Theory - A proof for us to do together. 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’) Cs173 - Spring 2004

  33. 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. CS 173 Set Theory - A proof for us to do together. Pv that if (A - B) U (B - A) = (A U B) then ______ A  B =  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. DeMorgan’s!! Then x is not in (A - B) U (B - A). Do you see the contradiction yet? But x is in A U B since (A  B)  (A U B). Thus, A  B = . Cs173 - Spring 2004