11 - Sets Introduction Discrete Mathematics Andrei Bulatov...

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Introduction Sets Discrete Mathematics Andrei Bulatov
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Discrete Mathematics - Sets 11-2 What is a Set A set is an unordered collection of objects This is not a rigorous definition!!! Every `conventional’ definition reduces the defined concept to a wider, more general, concept. For example, `A cow is a big animal with horns and four legs in the corners’ There is no more general concept than sets. Therefore a rigorous definition is impossible. If we use the `definition’ above, we get `naïve set theory’ Otherwise we need axiomatic set theory , Zermelo-Frenkel axioms (introduced by Skolem) ZF or ZFC.
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Discrete Mathematics - Sets 11-3 Elements, Describing a Set The objects in a set are called elements or members of the set One way to describe a set is to list its elements a A – a is an element of A, a belongs to A a A – a is not an element of A, a does not belong to A {0,1,2,3,4,5,6,7,8,9} – the set of digits {a,b,c,…,x,y,z} – set of Latin letters, alphabet A set can be an element of another set Set of alphabets: {{a,b,c,…}, { α , β , γ ,…},…}
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Discrete Mathematics - Sets 11-4 Set Builder Big sets can be described using set builder : {x | P(x)}, the set of all x such that P(x) {x | there is y such that x = 2y}, the set of even numbers = {x | 5 y (x =2y)} {x | x is a black cow} N = {0,1,2,3,…}, the set of natural numbers Z = {…,-2,-1,0,1,2,3,…}, the set of integers
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11 - Sets Introduction Discrete Mathematics Andrei Bulatov...

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