A relation is a set of ordered pairs. The set of first members of the ordered pairs is called the domain of the relation, and the set of second members is called the image. (Your teacher may call it the range, instead.) For example, in the relation {(1, 3), (2, 5)}, the domain is {1, 2} and the image is {3, 5}.

Some relations have the property that no first member appears more than once. They are called functions. For example, the relation above is a function. This agrees with the definition you're used to because a function tells us what to do with each input value. In the case of the function above, if x is 1 then y is 3, while if x is 2 then y is 5. (Most interesting functions, of course, have more than two legal values of x.) An example of a relation that is not a function is {(0, 1), (0, 2), (−1, 3)}.

Remember, the domain and image are both sets. That means it doesn't make any sense to list an element more than once. For example, the domain of the relation at the end of the last paragraph is just {0, −1}; we only write the 0 once.