Page 251 - HOW TO PROVE IT: A Structured Approach, Second Edition

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One-to-one and Onto 237
Note that our definition of one-to-one starts with the negation symbol ¬.
In other words, to say that f is one-to-one means that a certain situation does
not occur. The situation that must not occur is that there are two different
elements of the domain of f, a 1 and a 2 , such that f (a 1 ) = f (a 2 ). This situation
is illustrated in Figure 1(a). Thus, the function in Figure 1(a) is not one-to-one.
Figure 1(b) shows a function that is one-to-one.

Figure 1

If f : A → B, then to say that f is onto means that every element of B is the
image under f of some element of A. In other words, in the diagram of f, every
element of B has an edge pointing to it. Neither of the functions in Figure 1 is
onto, because in both cases there are elements of B without edges pointing to
them. Figure 2 shows two functions that are onto.

Figure 2