文档介绍：Chapter Three
Vector Functions
Relations and Functions
We begin with a review of the idea of a function. Suppose A and B are sets. The
Cartesian product A´ Bof these sets is the collection of all ordered pairs (a,b) such
that a Î A and b Î B . A relation R is simply a subset of A ´ B . The domain of R is
the set dom R = {a Î A:(a,b) Î R}. In case A = B and the domain of R is all of A, we call
R a relation on A. A relation R Ì A ´ B such that (a,b) Î R and (a,c) Î R only if b =
c is called a function. In other words, if R is a function, and a Î dom R , there is exactly
one ordered pair (a,b) Î R . The second “coordinate” b is thus uniquely determined by a.
It is usually denoted R(a) . If R Ì A ´ B is a relation, the inverse of R is the relation
R - 1 Ì B ´ A defined by R - 1 = {(b,a):(a,b) Î R} .
Example
Let A be the set of all people who have ever lived and let S Ì A ´ A be the relation
defined by S = {(a,b):b is the mother of a}. The S is a relation on A, and is, in fact, a
function. The relation S - 1 is not a function, and domS -1 ¹ A.
The fact that f Ì A ´ B is a function with dom f = A is frequently indicated by
writing f : A ® B , and we say f is a function from A to B. Very often a function f is
defined by specifying the domain, and giving a recipe for finding f(a). Thus we may
define the function f from the interval [0,1] to the real numbers b