Functions

Functions - Math 100 Functions Martin H. Weissman 1....

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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 high-school curriculum. There are many ways that functions are taught at the high-school 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 high-school 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.

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Functions - Math 100 Functions Martin H. Weissman 1....

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