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Unformatted text preview: Introduction Bijections and Cardinality Discrete Mathematics Andrei Bulatov 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. 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 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....
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- Spring '10