CS 173: Discrete Structures. Eric Shaffer Office Hour: Wed. 1-2, 2215 SC

Transcription

1 CS 173: Discrete Structures Eric Shaffer Office Hour: Wed. 1-2, 2215 SC

2 Agenda Sets (sections 2.1, 2.2) 2

3 Set Theory Sets you should know: Notation you should know: 3

4 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. Note: { } 4

5 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)). A B Venn Diagram 5

6 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

7 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)) A B 7

8 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}) Vacuously holds, because (x ) is false. Is {1,2,3}? No Is {,1,2,3}?! Yes! Is {,1,2,3}? Yes! 8

9 Quiz time: Set Theory - Definitions and notation Is {x} {x}? Yes Is {x} {x,{x}}? Is {x} {x,{x}}? Yes Yes Is {x} {x}? No 9

10 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. : and are read such that or where Ex. Let D(x,y) denote x is divisible by y. Give another name for { x : y ((y > 1) (y < x)) D(x,y) }. Primes Can we use any predicate P to define a set S = { x : P(x) }? 10

11 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 } No! Then, if S S, then by defn of S, S S. So S must not be in S, right? But, if S S, then by defn of S, S S. ARRRGH! There is a town with a barber who shaves all the people (and only the people) who don t shave themselves. Who shaves the barber? 11

12 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}, S = 3. If S = {3,3,3,3,3}, S = 1. If S =, S = 0. If S = {, { }, {,{ }} }, S = 3. If S = {0,1,2,3, }, S is infinite. (more on this later) 12

13 Set Theory - Power sets If S is a set, then the power set of S is 2 S = { x : x S }. aka P(S) If S = {a}, If S = {a,b}, If S =, 2 S = {, {a}}. We say, P(S) is the set of all subsets of S. If S = {,{ }}, 2 S = {, { }, {{ }}, {,{ }}}. Fact: if S is finite, 2 S = 2 S. (if S = n, 2 S = 2 n ) 13

14 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>} We ll use these special sets soon! A 1 x A 2 x x A n = {<a 1, a 2,, a n >: a 1 A 1, a 2 A 2,, a n A n } A,B finite AxB =? 14

15 Set Theory - Cartesian Product The Cartesian Product of two sets A and B is: A x B = { <a,b> : a A b B} A,B finite AxB =? 15

16 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} B A 16

17 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} B A 17

18 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 18

19 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} = B A Sets whose intersection is empty are called disjoint sets 19

20 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 = A = U and U = 20

21 The set difference, A - B, is: Set Theory - Operators U B A A - B = { x : x A x B } A - B = A B 21

22 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 U B A 22

23 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) 23

24 Set Theory - Famous Identities Page 124 Mostly like HS algebra. Don t memorize understand them 24

25 Set Theory - Famous Identities Identity A U = A A U = A Domination Idempotent A U U = U A = A A U A = A A A = A (Lazy) 25

26 Set Theory - Famous Identities Excluded Middle A U A = U Uniqueness A A = Double complement A = A 26

27 Set Theory - Famous Identities Commutativity A U B = A B = B U A B A Associativity (A U B) U C = (A B) C = A U (B U C) A (B C) Distributivity A U (B C) = A (B U C) = (A U B) (A U C) (A B) U (A C) 27

28 Set Theory - Famous Identities DeMorgan s I (A U B) = A B DeMorgan s II (A B) = A U B p q 28

29 Set Theory - 4 Ways to prove identities Show that A B and that A B. New & important Use a membership table. Like truth tables Use previously proven identities. Like Use logical equivalences to prove equivalent set definitions. Not hard, a little tedious 29

30 Set Theory - 4 Ways to prove identities Prove that (A U B) = A B using a membership table. 0 : x is not in the specified set 1 : otherwise A B A B A B A U B A U B Haven t we seen this before? 30

31 CS 173 Set Theory - 4 Ways to prove identities Prove that (A U B) = A B using identities. (A U B) = A U B = A B = A B 31

32 CS 173 Set Theory - 4 Ways to prove identities Prove that (A U B) = A B equivalent set definitions. using logically (A U B) = {x : (x A v x B)} = {x : (x A) (x B)} = {x : (x A) (x B)} = A B 32

33 Set Theory - 4 Ways to prove identities Prove that (A U B) = A B 1. ( ) (x A U B) 2. ( ) (x A B) 33

34 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 ) 34

35 Set Theory - A proof for us to do together. Pv that if (A - B) U (B - A) = (A U B) then A B = a) A U B = Suppose to the contrary, that A B, and that x A B. b) A = B c) A B = Then x cannot be in A-B and x cannot be in B-A. d) A-B = 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 =. Trying to pv p --> q Assume p and not q, and find a contradiction. Our contradiction was that sets weren t equal. 35