The basics

A two-place relation is a way of pairing up members of one set with members of another set. We can use a diagram to represent a relation; there is an arrow for each pair, going from the first item in the pair to the second. Thus all the figures in this chapter, starting with Figure 4.1, represent two-place relations.

If there is an arrow x → y in the relation, we say that “x maps to y” and that “y is the image of x” under the relation. Thus in the relation depicted by Figure 4.1, the elements 1, 4, and 5 all map to 96, 1 also maps to 94, the element 2 maps to 100, and finally 3 maps to 99. Another way to say the same thing is that 96 is the image of 1 and is also the image of 4 and of 5, and so forth.

A one-input function is a special kind of two-place relation, one for which each item in the first set has exactly one outgoing arrow, that is, each such item maps to exactly one element of the second set. Thus Figure 4.2 represents a one-input function, but Figure 4.1 does not. In fact, there are two ways in which the relation depicted in Figure 4.1 fails to be a function. There are elements of the first set (namely 1) that maps to two things, and there are elements (namely 6) that map to no elements.