{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}