CS2800-Sets_v.4

CS2800-Sets_v.4 - Discrete Math CS 2800 Prof Bart Selman...

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1 Discrete Math CS 2800 Prof. Bart Selman [email protected] Module Basic Structures: Sets Rosen, chapt. 2.
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2 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: { }
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3 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.
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4 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. Let D(x,y) denote “x is divisible by y.” Give another name for { x : 2200 y ((y > 1) (y < x)) ¬ D(x,y) }. Can we use any predicate P to define a set S = { x : P(x) }? “Any property should define a set… perhaps…” : and | are read “such that” or “where” Primes
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5 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. So S must not be in S, right? ARRRGH! But, if S S, then by defn of S, S S. No! 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? “the set of all sets that do not contain themselves as members” Reveals contradiction in Frege’s naïve set theory . Avoid self-reference. Use hierarchy of sets (types). Now, what about S itself? It’s a set. Is it in S?
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6 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. )) , ( ) , ( )( ( ) ( ( ) ( y x shaves y y shaves y x barber x ¬ 2200 5 This sentence is unsatisfiable (a contradiction) because of the universal quantifier. The universal quantifier y will include every single element in the domain,
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CS2800-Sets_v.4 - Discrete Math CS 2800 Prof Bart Selman...

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