The hierarchy of functions and relations

There are many different types of mathematical objects that we could study in this class. Some of them are relation-like objects:

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

some of them are function-like objects:

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

As you go along these lists, both the number of variables and the number of dimensions needed for graphing increase, as in the following diagram:

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. You can also use a variable to give a name to a truth value, so maybe p stands for {0 < 2}, although we won't need to do this.

A constant is, in this class, 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. Again, you can use a variable to stand for a constant, so maybe a stands for 2; in other words, a = −2.

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. If A stands for the set {x | x < 2}, then these two statements mean the same thing:

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

The first of these says that x belongs to the set A, while the second uses the definition of A to say exactly what that means.

A function, or unary function for emphasis, is a rule for taking a 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. If f stands for the function (x ↦ x − 2), then these two expressions mean the same thing:

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

The first of these is the value of the function f at the argument x, while the second uses the definition of f to say exactly what that means.

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. If R stands for the relation {x, y | x + y < 2}, then these two statements mean the same thing:

(x, y) ∈ R;
x + y < 2.

The first of these says that x and y are related by the relation R, while the second uses the definition of R to say exactly what that means.

A binary function, or function of two variables, is a rule for taking an ordered pair of two inputs and using it to calculate an output. An example is (x, y ↦ x + y − 2), the rule which subtracts 2 from the sum of the two inputs. To graph a binary function, we need three dimensions, two for the inputs and one for the output. If g stands for the function (x, y ↦ x + y − 2), then these two expressions mean the same thing:

g(x, y);
x + y − 2.

A ternary relation, or relation between three variables, is a set of ordered triples instead of a set of ordered pairs. An example is {x, y, z | x + y + z < 2}. We again use three dimensions to graph a ternary relation.

A ternary function, or function of three variables, is a rule for taking an ordered triple of three inputs and using it to calculate an output. An example is (x, y, z ↦ x + y + z − 2), the rule which subtracts 2 from the sum of the three inputs. To graph a ternary function, we need four dimensions, three for the inputs and one for the output.

A quaternary relation, or relation between four variables, is a set of ordered quadruples. An example is {x₁, x₂, x₃, x₄ | x₁ + x₂ + x₃ + x₄ < 2}. We again use four dimensions to graph a quaternary relation.

We can continue with quaternary functions, quinary functions, etc, which are functions of four or more variables; and we can continue with quinary relations, senary relations, etc, which are relations between five or more variables. (But around this point, most people stop using the ‘‑ary’ terms, because few people can remember them.)

There are various relationships between these different kinds of objects:

The domain of a function of n variables is a relation in n variables (the same n variables);
The range of a function of any number of variables is a set (a relation in 1 variable, the output).
The graph of a function of n variables is the graph of a relation in n + 1 variables (the n input variables plus the 1 output variable), which contains all of the information in the function.

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This web page was written between 2010 and 2016 by Toby Bartels, last edited on 2016 April 7. Toby reserves no legal rights to it. The diagram was drawn with the aid of Jacques Distler's SVG editor.

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