17a-Cardinality

17a-Cardinality - IBijection and Cardinality ntroduction...

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Bijection and Cardinality Discrete Mathematics

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Discrete Mathematics - Cardinality 17-2 Previous Lecture Functions Describing functions Injective functions Surjective functions Bijective functions
Discrete Mathematics - Cardinality 17-3 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.

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Discrete Mathematics - Cardinality 17-4 Composition of Functions Let g be a function from A to B and let f be a function from B to C. The composition of the functions f and g, denoted by f g, is the function from A to C defined by ( f g)(a) = f( g( a )) g f a g(a) f(g(a)) g(a) f(g(a)) (f g)(a) f g A B C
Discrete Mathematics - Cardinality 17-5 89 100 90 80 70 49 0 79 69 50 Composition of Functions (cntd) Suppose that the students first get numerical grades from 0 to 100 that are later converted into letter grade. Adams Chou Goodfriend Rodriguez Stevens A B C D F 100 95 92 84 75 40 0 Adams Chou Goodfriend Rodriguez Stevens B C D F Let f(a) = b mean `b is the father of a’. What is f f ? f g g f

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Discrete Mathematics - Cardinality 17-6 Composition of Numerical Functions Let g(x) = and f(x) = x + 1. Then (f g)(x) = f( g( x )) = g(x) + 1 = + 1 Thus, to find the composition of numerical functions f and g given by formulas we have to substitute g(x) instead of x in f(x).
Discrete Mathematics - Cardinality 17-7 Inverse Functions Let f be a one-to-one correspondence from the set A to the set B. The inverse function of f

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17a-Cardinality - IBijection and Cardinality ntroduction...

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