functions - NOTES ON FUNCTIONS These notes will cover some...

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Unformatted text preview: NOTES ON FUNCTIONS These notes will cover some terminology regarding functions not included in Solows book. You should read Appendix A.2 in the book before reading these notes. Definition 1. We say that two functions f and g are equal if they have the same domain and codomain, and f ( a ) = g ( a ) for all a in the domain. Note that we require the functions to have the same domain and codomain for them to be equal. For example, the functions f : R R and g : Z Z given by f ( x ) = x 2 and g ( x ) = x 2 are defined by the same formula, but they are not equal since they have different domains and codomains. The function h : R { x R | x } given by h ( x ) = x 2 again is defined by the same formula as f , and now has the same domain as f , but since they have different codomains, they are not equal. Definition 2. The identity function on a set A , denoted by id A , is the function from A to itself such that id A ( a ) = a for all a A . Definition 3. If f : A B and g : B C are functions, we define their composition , denoted by g f , to be the function g f : A C defined by ( g f )( a ) = g ( f ( a )). Definition 4. A function f : A B is called bijective if it is both injective and surjective. Again, we have to be careful about the domain and codomain on which a function is defined. Consider a function given by the formula f ( x ) = x 2 . It makes no sense to say this is injective, surjective, nor bijective without specifying what domain and codomain...
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functions - NOTES ON FUNCTIONS These notes will cover some...

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