Page 248 - HOW TO PROVE IT: A Structured Approach, Second Edition

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234 Functions
2
(b) Let f : R → R be the function defined by the formula f (x) = x −
2x. What is f (2)?
(c) Let f ={(x, n) ∈ R × Z | n ≤ x < n + 1}. Then f : R → Z. What
is f (π)? What is f (−π)?
4. (a) Let N be the set of all countries and C the set of all cities. Let H :
N → C be the function defined by the rule that for every country n,
H(n) = the capital of the country n. What is H(Italy)?
(b) Let A ={1, 2, 3} and B = P (A). Let F : B → B be the function
defined by the formula F(X) = A \ X. What is F({1, 3})?
(c) Let f : R → R × R be the function defined by the formula f (x) =
(x + 1, x − 1). What is f (2)?
∗ 5. Let L be the function defined in part 3 of Example 5.1.2 and let H be the
function defined in exercise 4(a). Describe L ◦ H and H ◦ L.
6. Let f and g be functions from R to R defined by the following formulas:
1
f (x) = , g(x) = 2x − 1.
2
x + 2
Find formulas for ( f ◦ g)(x) and (g ◦ f )(x).
7. Suppose f : A → B and C ⊆ A. The set f ∩ (C × B), which is a relation
∗
from C to B, is called the restriction of f to C, and is sometimes denoted
f C. In other words,
f C = f ∩ (C × B).
(a) Prove that f C is a function from C to B and that for all c ∈ C,
f (c) = ( f C)(c).
(b) Suppose g : C → B. Prove that g = f C iff g ⊆ f .
(c) Let g and h be the functions defined in parts 2 and 3 of Example 5.1.3.
Show that g = h Z.
8. Suppose A is a set. Show that i A is the only relation on A that is both an
equivalence relation on A and also a function from A to A.
9. Suppose f : A → C and g : B → C.
(a) Prove that if A and B are disjoint, then f ∪ g : A ∪ B → C.
(b) More generally, prove that f ∪ g : A ∪ B → C iff f (A ∩ B) =
g (A ∩ B).(Seeexercise7forthemeaningofthenotationusedhere.)
∗
10. Suppose R is a relation from A to B, S is a relation from B to C,
Ran(R) = Dom(S) = B, and S ◦ R : A → C.
(a) Prove that S : B → C.
(b) Give an example to show that it need not be the case that R : A → B.
11. Suppose f : A → B and S is a relation on B. Define a relation R on A
as follows:

R ={(x, y) ∈ A × A | ( f (x), f (y)) ∈ S}.