18h - Introduction Cardinality Discrete Mathematics Andrei...

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Unformatted text preview: Introduction Cardinality Discrete Mathematics Andrei Bulatov Discrete Mathematics - Cardinality 18-2 Properties of Functions A function f is said to be one-to-one , or injective , if and only if f(a) = f(b) implies a = b. A function f from A to B is called onto , or surjective , if and only if for every element b ∈ B there is an element a ∈ A with f(a) = b. A function is called a surjection if it is onto. A function f is a one-to-one correspondence , or a bijection , if it is both one-to-one and onto. Discrete Mathematics - Cardinality 18-3 How to Count Elements in a Set How many elements are in a set? Easy for finite sets, just count the elements. What about infinite sets? Does it make sense at all to ask about the number of elements in an infinite set? Can we say that this infinite set is larger than that infinite set? Which set is larger: the set of all integers or the set of even integers? the set of all integers or the set of all rationals? the set of all integers or the set of all reals? Discrete Mathematics - Cardinality 18-4 Cardinality and Bijections If A and B are finite sets, it is not hard to see that they have the same cardinality if and only if there is a bijection from A to B For example, |{a,b,c}| = |{d,e,f}| a b c d e f { } { } a b c d e f Discrete Mathematics - Cardinality 18-5 Cardinality and Bijections Sets A and B (finite or infinite) have the same cardinality if and only if there is a bijection from A to...
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18h - Introduction Cardinality Discrete Mathematics Andrei...

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