MATH 092 Lecture 7: FUNCTIONS AND MAPPINGS

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1 Aug 2016
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FUNCTIONS AND MAPPINGS
A Junction is a relation J such that each domain element x is paired with exactly
one range element y. This property can be expressed as follows:
-<x, y> EJ and -<x, Z> EJ =} y = z.
The y which is thus uniquely determined by f and x is designated f(x):
y = f(x) ~ <x, y> Ef.
One tends to think of a function as being active and a relation which is not
a function as being passive. A function f acts on an element x in its domain to
givef(x). We take x and apply fto it; indeed we often call a function an operator.
On the other hand, if R is a relation but not a function, then there is in general
no particular y related to an element x in its domain, and the pairing of x and y
is viewed more passively.
We often define a function f by specifying its value f(x) for each x in its
domain, and in this connection a stopped arrow notation is used to indicate the
pairing. Thus x 1-+ x2 is the function assigning to each number x its square x2
If we want it to be understood that f is this function, we can write "Consider
the function The domain of f must be understood for this notation
to be meaningful.
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