Page 241 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Functions 227
6. Let A be any set. Recall that i A ={(a, a) | a ∈ A} is called the identity
relation on A. Is it a function from A to A?
2
7. Let f ={(x, y) ∈ R × R | y = x }.Is f a function from R to R?
Solutions
1. Yes. Note that 1 is paired with 5 in the relation F, but it is not paired with
any other element of B. Similarly, 2 is paired only with 4, and 3 with 5. In
other words, each element of A appears as the first coordinate of exactly
one ordered pair in F. Therefore F is a function from A to B. Note that the
definition of function does not require that each element of B be paired with
exactly one element of A. Thus, it doesn’t matter that 5 occurs as the second
coordinate of two different pairs in F and that 6 doesn’t occur in any ordered
pairs at all.
2. No. G fails to be a function from A to B for two reasons. First of all, 3
isn’t paired with any element of B in the relation G, which violates the
requirement that every element of A must be paired with some element of
B. Second, 1 is paired with two different elements of B, 5 and 6, which
violates the requirement that each element of A be paired with only one
element of B.
3. If we make the reasonable assumption that every city is in exactly one
country, then L is a function from C to N.
4. Because some people have no children and some people have more than
one child, C is not a function from P to P.
5. Yes. D is a function from P to P (P). Each person p is paired with exactly
one set x ⊆ P, namely the set of all children of p. Note that in the relation
D, a person p is paired with the set consisting of all of p’s children, not with
the children themselves. Even if p does not have exactly one child, it is still
true that there is exactly one set that contains precisely the children of p and
nothing else.
6. Yes. Each a ∈ A is paired in the relation i A with exactly one element of A,

namely a itself. In other words, (a, a) ∈ i A , but for every a = a, (a, a ) /∈
i A . Thus, we can call i A the identity function on A.
2
7. Yes. For each real number x there is exactly one value of y, namely y = x ,
such that (x, y) ∈ f .
Suppose f : A → B.If a ∈ A, then we know that there is exactly one b ∈ B
such that (a, b) ∈ f . This unique b is called “the value of f at a,” or “the image
of a under f,” or “the result of applying f to a,” or just “f of a,” and it is written
f (a). In other words, for every a ∈ A and b ∈ B, b = f (a)iff (a, b) ∈ f .