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CS 173: Discrete Mathematical Structures

CS 173: Discrete Mathematical Structures. Cinda Heeren heeren@cs.uiuc.edu Rm 2213 Siebel Center Office Hours: W 12:30-2:30p. What homework? I’ve read it. 25% done 50% done 75% done. CS 173 Announcements. Homework 2 returned this week. Homework 3 available. Due 09/16.

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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 12:30-2:30p

  2. What homework? • I’ve read it. • 25% done • 50% done • 75% done CS 173 Announcements • Homework 2 returned this week. • Homework 3 available. Due 09/16. • LaTex workshop Th, 9/15, at 6:30p in SC 2405. • The following reflects my status on hwk #3… Cs173 - Spring 2004

  3. |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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. U A B 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

  13. 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

  14. 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

  15. 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

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

  17. 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

  18. (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

  19. 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

  20. (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

  21. (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

  22. (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

  23. (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

  24. 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’) = X  (Y - Z) Cs173 - Spring 2004

  25. 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

  26. CS 173 Set Theory - Generalized Union Ex. Let U = N, and define: A1 = {2,3,4,…} A2 = {2,4,6,…} A3 = {3,6,9,…} Cs173 - Spring 2004

  27. Primes Composites  N I have no clue. primes CS 173 Set Theory - Generalized Union Ex. Let U = N, and define: Then Cs173 - Spring 2004

  28. CS 173 Set Theory - Generalized Intersection Ex. Let U = N, and define: A1 = {1,2,3,4,…} A2 = {2,4,6,…} A3 = {3,6,9,…} Cs173 - Spring 2004

  29. Multiples of LCM(1,…,n) CS 173 Set Theory - Generalized Intersection Ex. Let U = N, and define: Then Cs173 - Spring 2004

  30. B A Wrong. CS 173 Set Theory - Inclusion/Exclusion Example: How many people are wearing a watch? How many people are wearing sneakers? How many people are wearing a watch OR sneakers? What’s wrong? |A  B| = |A| + |B| - |A  B| Cs173 - Spring 2004

  31. 125 173 217 - (157 + 145 - 98) = 13 CS 173 Set Theory - Inclusion/Exclusion Example: There are 217 cs majors. 157 are taking cs125. 145 are taking cs173. 98 are taking both. How many are taking neither? Cs173 - Spring 2004

  32. Now let’s do it for 4 sets! kidding. CS 173 Set Theory - Generalized Inclusion/Exclusion Suppose we have: B A C And I want to know |A U B U C| |A U B U C| = |A| + |B| + |C| - |A  B| - |A  C| - |B  C| + |A  B  C| Cs173 - Spring 2004

  33. CS 173 Set Theory - Inclusion/Exclusion Example: How many people are wearing a watch? How many people are wearing sneakers? How many people are wearing a watch AND sneakers? How many people are wearing a watch OR sneakers? Cs173 - Spring 2004

  34. CS 173 Set Theory - Generalized Inclusion/Exclusion For sets A1, A2,…An we have: Cs173 - Spring 2004

  35. (101011) CS 173 Set Theory - Sets as bit strings Let U = {x1, x2,…, xn}, and let A  U. Then the characteristic vector of A is the n-vector whose elements, xi, are 1 if xi A, and 0 otherwise. Ex. If U = {x1, x2, x3, x4, x5, x6}, and A = {x1, x3, x5, x6}, then the characteristic vector of A is Cs173 - Spring 2004

  36. Bit-wise OR Bit-wise AND CS 173 Set Theory - Sets as bit strings Ex. If U = {x1, x2, x3, x4, x5, x6}, A = {x1, x3, x5, x6}, and B = {x2, x3, x6}, Then we have a quick way of finding the characteristic vectors of A  B and A  B. Cs173 - Spring 2004

  37. f(x) = -(1/2)x - 25 domain co-domain CS 173 Functions Suppose we have: And I ask you to describe the yellow function. What’s a function? Notation: f: RR, f(x) = -(1/2)x - 25 Cs173 - Spring 2004

  38. CS 173 Functions Definition: a function f : A  B is a subset of AxB where  a  A, ! b  B and <a,b>  f. Cs173 - Spring 2004

  39. B A A point! A collection of points! CS 173 Functions Definition: a function f : A  B is a subset of AxB where  a  A, ! b  B and <a,b>  f. B A Cs173 - Spring 2004

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