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Discrete Structures & Algorithms Basics of Set Theory EECE 320 — UBC

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2 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: { }

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3 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.” Venn Diagram or, “B contains A.” or, “every element of A is also in B.” or, x ((x A) (x B)). A B

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4 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

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5 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

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6 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! Vacuously

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7 Definitions and notation Quiz Time Is {x} {x}? Is {x} {x,{x}}? Is {x} {x,{x}}? Is {x} {x}? Yes No

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8 How to specify 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. Example: 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) }? : and | are read “such that” or “where” Primes

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9 Predicates for defining sets 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 the definition of S, S S. So S must not be in S, right? Doh! But, if S S, then by the definition of S, S S. No! There is a town with a barber who shaves all the people (and only the people) who do not shave themselves. Who shaves the barber?

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10 Cardinality of sets 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}, If S = , If S = { , { }, { ,{ }} }, |S| = 1. |S| = 0. |S| = 3. If S = {0,1,2,3,…}, |S| is infinite. (more on this later)

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

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12 Cartesian product The Cartesian product of two sets A and B is: A x B = { : a A b B} If A = {Charlie, Lucy, Linus}, and B = {Brown, VanPelt}, then A,B finite |AxB| = ? A 1 x A 2 x … x A n = { : a 1 A 1, a 2 A 2, …, a n A n } A x B = {,,,,, } We’ll use these special sets soon! a)AxB b)|A|+|B| c)|A+B| d)|A||B|

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13 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} A B

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14 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} A B

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15 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} A B

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

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17 Operators The complement of a set A is A c = { x : x A} If A = {x : x is bored}, then A = {x : x is not bored} A = c = U and U c = U

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18 Operators The set difference, A - B, is: A U B A - B = { x : x A x B } A - B = A B c

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19 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” A U B

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20 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)

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21 Famous identities Two pages of (almost) obvious equivalences. One page of HS algebra. Some new material? Don’t memorize them, understand them! They’re in Rosen, Sec. 2.2

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22 Identities Identity Domination Idempotent A U = A A U = A A U U = U A = A U A = A A A = A

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23 Identities Excluded middle Uniqueness Double complement A U A = U A A = A = A

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24 Identities Commutativity Associativity Distributivity A U B = (A U B) U C = A B = B U A B A (A B) C = A U (B U C) A (B C) A U (B C) = A (B U C) = (A U B) (A U C) (A B) U (A C)

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25 Identities DeMorgan’s Law I DeMorgan’s Law II pq Hand waving is good for intuition, but we aim for a more formal proof. (A U B) c = A c B c (A B) c = A c U B c

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26 Proving 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. New & importantLike truth tables Like Not hard, a little tedious

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27 Proving identities (1) Prove that 1.( ) (x (A U B) c ) (x A U B) (x A and x B) (x A c B c ) 2.( ) (x A c B c ) (x A and x B) (x A U B) (x (A U B) c ) (A U B) c = A c B c

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28 Proving identities (2) Prove that (A U B) c = A c B c using a membership table. 0 : x is not in the specified set 1 : otherwise ABAB A B A U B 1100010 1001010 0110010 0011101 Have we not seen this before?

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29 Proving identities (3) Prove that (A U B) c = A c B c using identities. (A U B) = A U B = A B

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30 Proving identities (4) Prove that (A U B) c = A c B c using logically equivalent set definitions. (A U B) c = {x : (x A v x B)} = {x : (x A) (x B)} = A c B c = {x : (x A c ) (x B c )}

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31 A proof to do X (Y - Z) = (X Y) - (X Z). True or False? Prove your response. = (X Y) (X’ U Z’) = (X Y X’) U (X Y Z’) = U (X Y Z’) = (X Y Z’) (X Y) - (X Z) = (X Y) (X Z)’

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32 Suppose to the contrary, that A B , and that x A B. Another proof to do Prove that if (A - B) U (B - A) = (A U B) then ______ Then x cannot be in A-B and x cannot be in B-A. Do you see the contradiction yet? But x is in A U B since (A B) (A U B). A B = Thus, A B = . a)A U B = b)A = B c)A B = d)A-B = B-A = Then x is not in (A - B) U (B - A). DeMorgan’s Law Trying to prove p q Assume p and not q, and find a contradiction. Our contradiction was that sets were not equal.

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33 Wrap up Sets are an essential structure for all mathematics. We covered the basic definitions and identities that will allow us to reason about sets.

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Sets Defined A set is an object defined as a collection of other distinct objects, known as elements of the set The elements of a set can be anything:

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