Functions of multiple variables

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 should evaluate to true or false.

A constant is, in this class, usually just a real number. 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 this class, 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 is a rule for taking one number (the input, or argument) and using it to calculate a number (the output, or value). 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 is a set of ordered pairs instead of a set of individual numbers. An example is {x,y | x + y < 2}.

A binary function or function of two variables is a rule for taking an ordered pair of two arguments 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 arguments. To graph a binary function, we need three dimensions, two for the input and one for the output.

We can continue with ternary functions, quaternary 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.

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This web page was written between 2010 and 2013 by Toby Bartels, last edited on 2013 January 24. Toby reserves no legal rights to it. The diagram was drawn with the aid of Jacques Distler's SVG editor.

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