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236 Functions
(b) Let S ={( f, g) ∈ F × F | f ∈ O(g)}. Prove that S is a preorder, but
not a partial order. (See exercise 24 of Section 4.6 for the definition
of preorder.)
(c) Suppose f 1 ∈ O(g) and f 2 ∈ O(g), and s and t are real numbers. De-
fine a function f : Z → R by the formula f (x) = sf 1 (x) + tf 2 (x).
+
Prove that f ∈ O(g). (Hint: You may find the triangle inequality
helpful. See exercise 12(c) of Section 3.5.)
17. (a) Suppose g : A → B and let R ={(x, y) ∈ A × A | g(x) = g(y)}.
Show that R is an equivalence relation on A.
(b) Suppose R is an equivalence relation on A and let g : A → A/R be
the function defined by the formula g(x) = [x] R . Show that R =
{(x, y) ∈ A × A | g(x) = g(y)}.
∗
18. Suppose f : A → B and R is an equivalence relation on A. We will
say that f is compatible with R if ∀x ∈ A∀y ∈ A(xRy → f (x) = f (y)).
(You might want to compare this exercise to exercise 23 of Section 4.6.)
(a) Suppose f is compatible with R. Prove that there is a unique function
h : A/R → B such that for all x ∈ A, h([x] R ) = f (x).
(b) Suppose h : A/R → B and for all x ∈ A, h([x] R ) = f (x). Prove that
f is compatible with R.
19. Let R ={(x, y) ∈ Z × Z | x ≡ y (mod 5)}. Recall that we saw in Sec-
tion 4.6 that R is an equivalence relation on Z.
(a) Show that there is a unique function h : Z/R → Z/R such that for
2
every integer x, h([x] R ) = [x ] R . (Hint: Use exercise 18.)
(b) Show that there is no function h : Z/R → Z/R such that for every
x
integer x, h([x] R ) = [2 ] R .

5.2. One-to-one and Onto

In the last section we saw that the composition of two functions is again a
function. What about inverses of functions? If f : A → B, then f is a relation
from A to B,so f −1 is a relation from B to A. Is it a function from B to A? We’ll
answer this question in the next section. As we will see, the answer hinges on
the following two properties of functions.
Definition 5.2.1. Suppose f : A → B. We will say that f is one-to-one if
¬∃a 1 ∈ A∃a 2 ∈ A( f (a 1 ) = f (a 2 ) ∧ a 1 = a 2 ).
We say that f is onto if
∀b ∈ B∃a ∈ A( f (a) = b).
One-to-one functions are sometimes also called injections, and onto functions
are sometimes called surjections.