##
*
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.
5 ---------- begin topic:

Link Information

A More General Definition of the word *Function* **:**