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# functionskey - McCombs Math 381 Working with Functions...

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McCombs Math 381 Working with Functions 1 Given sets A and B , with a relation f from A to B . Important Definitions: 1. f : A B is a function from A to B provided the following are both true: (i) ( ) b a f B b A a = : , (ii) If ( ) ( ) , and c a f b a f = = then b = c . Given a function f : A B . 2. The domain of f is given by dom f = A . The codomain of f is given by codom f B = . 3. The image of f is given by ( ) ( ) { } b a f A a B b A f f = = = with : im . Note that image of f is also called the range of f 4. f is called a one-to-one ( injective ) function provided A a a 2 1 , , if 2 1 a a , then ( ) ( ) 2 1 a f a f 5. f is called an onto ( surjective ) function provided ( ) b a f A a B b = : , In other words, ( ) B A f f = = im . 6. f is called a bijection provided f one-to-one and onto. Important Propositions: 1. Given f : A B , the inverse relation f 1 : B A is a function if and only if f is a bijection . 2. Given finite sets A and B , with a A = and b B = . (i) The total number of functions from A to B , is given by b a . (ii) If B A > , there are no injections from A to B . (iii) If B A < , there are no surjections from A to B . (iv) If B A , the total number of injections from A to B is given by ( ) ! ! b b a . (v) If B A = , the total number of bijections from A to B is given by ! a .

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