- truth values,
- sets,
- relations,
- ternary relations,
- etc;

- constants,
- functions,
- binary functions,
- ternary functions,
- etc.

A **truth value** is either true or false;
any statement with no variables in it,
such as the statement that 0 < 2,
should evaluate to true or false (in this case, true).
To indicate that you are talking about the truth value of this statement,
rather than asserting the statement itself,
{0 < 2} is the truth value that 0 is less than 2
(which is the true truth value rather than the false one);
there are other notations used for this.
The graph of the true truth value is a solid dot,
while the graph of the false truth value is a hollow circle;
either way, this takes zero dimensions.

A **constant** is, in this class, usually a *real number*,
such as −2.
Any expression with no variables should evaluate to a constant,
but we use one dimension to graph a constant on a number line.

A **set** is, in the simplest case,
a *set of real numbers*.
A statement with one variable defines a set,
such as {*x* | *x* < 2},
the set of real numbers that are less than 2.
We again use one dimension to graph a set.

A **function**, or *unary function* for emphasis,
is a rule
for taking one number (the *input*)
and using it to calculate a number (the *output*).
An example is (*x* ↦ *x* − 2),
the rule which subtracts 2 from any number.
To graph a function, we need two dimensions,
one for the input and one for the output.

A **relation**, or *binary relation* for emphasis,
is a set of ordered pairs instead of a set of individual numbers.
An example
is
{*x*, *y* | *x* + *y* < 2}.
We again use two dimensions to graph a relation.

We can continue with binary functions, ternary functions, etc, which take two or more numbers as inputs; and we can continue with ternary relations, quaternary relations, etc, which relate three or more numbers. But we will not actually study these in this class.

*x*∈*A*, usually pronounced ‘*x*in*A*’;*x*< 2.

*f*(*x*), usually pronounced ‘*f*of*x*’;*x*− 2.

One of the basic principles of the theory of functions is that the only information necessary to specify a function is to show how to calculate its value at any argument. Therefore, if I write

and state that this holds forf(x) =x− 2

Or we can say thatf(5) = (5) − 2 = 3.

Notice that I always put parentheses around an expression when I substitute it for a variable; in this case, it wasn't really necessary, but it's best to play it safe.f(2x+ 3) = (2x+ 3) − 2 = 2x+ 1.

- {
*x*,*y*|*y*=*f*(*x*)}.

- A graph of a relation is the graph of a function if and only if every vertical line goes through the graph at most once.

Sometimes a vertical line doesn't go through the graph at all!
This happens when *f*(*x*) is undefined.
The **domain** of *f*
is the set of all inputs where *f* is defined:

- dom
*f*= {*x*|*f*(*x*) exists}.

Go back to the course homepage.

This web page was written between 2010 and 2014 by Toby Bartels, last edited on 2014 April 16. Toby reserves no legal rights to it. The diagram was drawn with the aid of Jacques Distler's SVG editor.

The permanent URI of this web page
is
`http://tobybartels.name/MATH-1150/2015SU/functions/`

.