- 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
(possibly undefined),
and 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}.

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