Function from X to Y

The essential idea
in a function from one set, X, to another set, Y, is that
for every element, x, in X
there is associated in some way
one and only one element of Y,
which is denoted by f(x).

We refer to the function in symbols as follows:

The x in the expression f(x) is often called the argument of the function.
The argument is whatever appears inside of the parenthesis.
f(x) is the value of the function at x.

A formal definition is the following:

A function from X to Y,
that is
is any subset of the cartesian product of X and Y which has the property that
each element of X appears in one and only one of the ordered pairs in .
Thus, for each element of X there is associated one and only one element of Y.

See ita-1-3-4-pg8.

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one-one functions

A function is called one-to-one, or one-one,
if different elements of X correspond under f to different elements of Y.

See ita-1-3-7-pg10.

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onto functions

A function is called onto
if each element of Y corresponds under f to some element of X.

See ita-1-3-7-pg10.

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one-one correspondence

If is both one-one and onto
it is called one-one onto or a one-one correspondence between X and Y.

See ita-1-3-7-pg10.

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Link Information
A More General Definition of the word Function :