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Unformatted text preview: Math 100 Functions Martin H. Weissman 1. Definitions The concept of function is a core concept in the highschool curriculum. There are many ways that functions are taught at the highschool level: Functions are graphs which pass the vertical line test. Functions are machines which output one number for each number input. Functions are expressions involving a bunch of operations and constants, and one letter, called x . In advanced mathematics, we use a definition of function which is based on set theory. The abstract definition is most closely related to the first highschool definition the vertical line test though the abstract definition is less obviously geometric. Let A and B be sets. Then A B (called the Cartesian product of A and B ) is the set of ordered pairs, whose first entry is an element of A , and whose second entry is an element of B . For example, if A = { a,b,c } , and B = { a,d } , then the set A B has the following six elements: A B = { ( a,a ) , ( a,d ) , ( b,a ) , ( b,d ) , ( c,a ) , ( c,d ) } . In general, if A and B are finite sets, and A has x elements, and B has y elements, then A B is a set with x y elements. Hopefully, this justifies the notation! Definition 1. A function from A to B is a subset f A B , such that the following condition holds: For all a A , there exists b B , such that ( a,b ) f . For all a A , for all b B , and for all b B , if ( a,b ) f and ( a,b ) f , then b = b ....
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This note was uploaded on 04/17/2008 for the course MATH 100 taught by Professor Weissman during the Fall '08 term at UCSC.
 Fall '08
 Weissman
 Math

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