notes-110327

notes-110327 - MATH 311-02 1 Notes Introduction to Higher...

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MATH 311-02 Notes Introduction to Higher Math 1 Function Attributes and Properties If a function is defined as f : A B , then the set A is called the domain of f ; the set B is the codomain of f . We might be curious about which elements of B are actually achieved by f ; i.e., which y B are such that for some x in A , f ( x ) = y ? We might call this set the image or range . Put in glorious set-builder-and-logic notation, the image of a function f : A B is { y B : x A, f ( x ) = y } We might look at a simple example: consider a function f : { 1 , 2 , 3 , 4 , 5 } → { 1 , 2 , 3 } such that f (1) = 2, f (2) = 3, f (3) = 3, f (4) = 2, and f (5) = 2. Alternatively, we might describe f as { (1 , 2) , (2 , 3) , (3 , 3) , (4 , 2) , (5 , 2) } . Then, as stated in the declaration of f , f has domain { 1 , 2 , 3 , 4 , 5 } and codomain { 1 , 2 , 3 } . The image of f is the set of second co-ordinates actually appearing in f , which is { 2 , 3 } — note that this is not quite the entire codomain. Here’s a more complicated example. We might let g : N → P ( N ); that is, g is a function mapping natural numbers to sets of natural numbers. We will define
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