Discrete Math CS 2800 Prof Bart Selman selmancs

Discrete Math CS 2800 Prof. Bart Selman selman@cs. cornell. edu Module Basic Structures: Sets Rosen, chapt. 2. 1

Set Theory - Definitions and notation A set is an unordered collection of objects referred to as elements. A set is said to contain its elements. Different ways of describing a set. 1 – Explicitly: listing the elements of a set {1, 2, 3} is the set containing “ 1” and “ 2” and “ 3. ” list the members between braces. {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, …, 99} is the set of positive integers less than 100; use ellipses when the general pattern of the elements is obvious. {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: { } 2

Set Theory - Definitions and notation 2 – Implicitly: by using a set builder notations, stating the property or properties of the elements of the set. S = {m| 2 ≤ m ≤ 100, m is integer} S is the set of all m such that m is between 2 and 100 and m is integer. 3

Set Theory - Ways to define sets : and | are read “such that” or “where” 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. 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) }? “Any property should define a set… perhaps…” 4

Reveals contradiction in Frege’s naïve set theory. Avoid self-reference. Use hierarchy of sets (types). Set Theory - Russell’s Paradox “the set of all sets that do not contain themselves as members” Can we use any predicate P to define a set S = { x : P(x) }? No! Define S = { x : x is a set where x x } Now, what about S itself? It’s a set. So S must not be Is it in S? in S, right? Then, if S S, then by defn of S, S S. But, if S S, then by defn of S, S S. ARRRGH! Compare: 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? 5

Aside: this layman’s version of Russell’s paradox; has some drawbacks. The Barber Paradox 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? Does the barber shave himself? If the barber does not shave himself, he must abide by the rule and shave himself. If he does shave himself, according to the rule he will not shave himself. This sentence is unsatisfiable (a contradiction) because of the universal quantifier. The universal quantifier y will include every single element in the domain, including our infamous barber x. So when the value x is assigned to y, the sentence can be rewritten to: Contradiction! Logically inconsistent definition / description. Town cannot exist! 6

Set Theory - Definitions and notation Important Sets N = {0, 1, 2, 3, …}, the set of natural numbers, non negative integers. Z = { …, -2, -1, 0, 1, 2, 3, …), the set of integers Z+ = {1, 2, 3, …} set of positive integers Q = {p/q | p Z, q Z, and q 0}, set of rational numbers R, the set of real numbers Note: Real number are the numbers that can be represented by an infinite decimal representation, such as 3. 4871773339…. The real numbers include both rational, and irrational numbers such as π and the and can be represented as points along an infinitely long number line. 7

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)). Venn Diagram A B 8

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 9

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)) A B 10

Set Theory - Definitions and notation Quick examples: {1, 2, 3} {1, 2, 3, 4, 5} Is {1, 2, 3}? Yes! x (x ) (x {1, 2, 3}) holds (for all over empty domain) Is {1, 2, 3}? No! Is { , 1, 2, 3}? Yes! 11

Set Theory - Definitions and notation A few more: Is {a}? Yes Is {a} {a, {a}}? Yes Is {a}? No 12

Set Theory - Cardinality If S is finite, then the cardinality of S, |S|, is the number of distinct elements in S. |S| = 3. If S = {1, 2, 3}, If S = {3, 3, 3}, If S = , |S| = 1. |S| = 0. If S = { , { }, { }} }, |S| = 3. If S = {0, 1, 2, 3, …}, |S| is infinite. 13

Set Theory - Power sets If S is a set, then the power set of S is 2 S = { x : x S }. If S = {a}, 2 S = { , {a}}. 2 S = { , {a}, {b}, {a, b}}. If S = {a, b}, If S = , aka P(S) We say, “P(S) is the set of all subsets of S. ” 2 S = { }. If S = { , { }}, 2 S = { , { }, {{ }}, { }}}. Fact: if S is finite, |2 S| = 2|S|. (if |S| = n, |2 S| = 2 n) Why? 14

Set Theory – Ordered Tuples Cartesian Product When order matters, we use ordered n-tuples 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, Van. Pelt}, then A x B = {<Charlie, Brown>, <Lucy, Brown>, <Linus, Brown>, <Charlie, Van. Pelt>, <Lucy, Van. Pelt>, <Linus, Van. Pelt>} We’ll use these special sets soon! a) b) A 1 x A 2 x … x An = {<a 1, a 2, …, an>: a 1 A 1, a 2 A 2, …, an An} c) d) Size? n A, B finite |Ax. B| = ? n Ax. B |A|+|B| |A+B| |A||B| 15

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

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

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

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

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= “I. e. , A = U – A, where U is the universal set. A set fixed within the framework of a theory and consisting of all objects considered in theory. “ 20

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

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 A B 22

Set Theory – Operators Example proof 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) Q. E. D. 23

Directly from defns. Identity Domination Idempotent Set Theory - Identities A U=A AUU=U A =A AUA=A A A=A 24

Set Theory – Identities, cont. Complement Laws AUA=U A A= Double complement A=A 25

Set Theory - Identities, cont. Commutativity Associativity Distributivity AUB= BUA A B= B A (A U B) U C = A U (B U C) (A B) C = A (B C) A U (B C) = (A U B) (A U C) A (B U C) = (A B) U (A C) 26

De. Morgan’s II (A U B) = A B Compare to rules for logical connectives. There is a deep connection via Boolean algebra. (A B) = A U B A B Proof by “diagram” (useful!), but we aim for a more formal proof. 27

Proving identities Prove that (A U B) = A B 1. ( ) (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) (De Morgan) 28

A Alt. proof B Prove that(A U B) = A B using a membership table. 0 : element x is not in the specified set 1 : otherwise A B A B AUB 1 1 0 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 Haven’t we seen this before? General connection via Boolean algebras (Rosen chapt. 11)

Note: contrast with earlier proof! (via set membership; subsets) Careful with “=“ in proof. Make sure is true equivalence. Proof using logically equivalent set definitions: (A U B) = A B (A U B) = {x : (x A v x B)} = {x : (x A) (x B)} = {x : (x A) (x B)} =A B 30

Example X (Y - Z) = (X Y) - (X Z). True or False? Proof: (X Y) - (X Z) = (X Y) (X Z)’ = (X Y) (X’ U Z’) = (X Y X’) U (X Y Z’) = U by defn. of - by de Morgan by distributive law (X Y Z’) = X (Y - Z) by defn. of - Note: Z’ = Z 31

Example A B= Pove that if (A - B) U (B - A) = (A U B) then ______ Proof by contradiction. 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. Then x is not in (A - B) U (B - A). a) A = B b) A B = c) A-B = B-A = Do you see the contradiction yet? But x is in A U B since (A B) (A U B). Thus, (A - B) U (B - A) ≠ (A U B). Contradiction. Thus, A B = . Trying to prove p --> q Assume p and not q, and find a contradiction. Our contradiction was that sets weren’t equal.

Set Theory - Generalized Union/Intersection 34

Set Theory - Generalized Union/Intersection Ex. Suppose that: 35

Example: Ex. Let U = N, and define: i=1, 2, …, N A 1 = {2, 3, 4, …} primes A 2 = {4, 6, 8, …} A 3 = {6, 9, 12, …} Note: i starts at 2 Union is all the composite numbers. a) b) c) d) e) Primes Composites N I have no clue.

Set Theory - Inclusion/Exclusion Example: There are 217 cs majors. 157 are taking cs 2800. 145 are taking cs 2110. 98 are taking both. 157 145 How many are taking neither? 217 - (157 + 145 - 98) = 13 37

Consider three sets A, B, and C. B A C And I want to know |A U B U C| What about 4 sets? |A U B U C| = |A| + |B| + |C| - |A B| - |A C| - |B C| + |A B C| Why final term?

Generalized Inclusion/Exclusion For sets A 1, A 2, …An we have:

Set Theory - Sets as bit strings Let U = {x 1, x 2, …, 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 = {x 1, x 2, x 3, x 4, x 5, x 6}, and A = {x 1, x 3, x 5, x 6}, then the characteristic vector of A is (101011)

Sets as bit strings Ex. If U = {x 1, x 2, x 3, x 4, x 5, x 6}, A = {x 1, x 3, x 5, x 6}, and B = {x 2, x 3, x 6}, Then we have a quick way of finding the characteristic vectors of A B and A B. Bit-wise OR Bit-wise AND A 1 0 1 1 B 0 1 1 0 0 1 A B 1 1 A B 0 0 1 1 1 0 0 1 41