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Functions

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

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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 0 B , if ( a, b ) f and ( a, b 0 ) f , then b = b 0 .

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