Page 249 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Functions 235
(a) Prove that if S is reflexive, then so is R.
(b) Prove that if S is symmetric, then so is R.
(c) Prove that if S is transitive, then so is R.
12. Suppose f : A → B and R is a relation on A. Define a relation S on B
∗
as follows:

S ={(x, y) ∈ B × B |∃u ∈ A∃v ∈ A( f (u) = x ∧ f (v) = y ∧ (u,v) ∈ R)}.

Justify your answers to the following questions with either proofs or
counterexamples.
(a) If R is reflexive, must it be the case that S is reflexive?
(b) If R is symmetric, must it be the case that S is symmetric?
(c) If R is transitive, must it be the case that S is transitive?
13. Suppose A and B are sets, and let F ={ f | f : A → B}. Also, suppose
R is a relation on B, and define a relation S on F as follows:

S ={( f, g) ∈ F × F |∀x ∈ A(( f (x), g(x)) ∈ R)}.

Justify your answers to the following questions with either proofs or
counterexamples.
(a) If R is reflexive, must it be the case that S is reflexive?
(b) If R is symmetric, must it be the case that S is symmetric?
(c) If R is transitive, must it be the case that S is transitive?
14. Suppose A is a nonempty set and f : A → A.
(a) Suppose there is some a ∈ A such that ∀x ∈ A( f (x) = a). (In this
case, f is called a constant function.) Prove that for all g : A → A,
f ◦ g = f .
(b) Suppose that for all g : A → A, f ◦ g = f . Prove that f is a constant
function. (Hint: What happens if g is a constant function?)
15. Let F ={ f | f : R → R}. Let R ={( f, g) ∈ F × F |∃a ∈ R∀x >
a( f (x) = g(x))}.
(a) Let f : R → R and g : R → R be the functions defined by the for-
mulas f (x) =|x| and g(x) = x. Show that ( f, g) ∈ R.
(b) Prove that R is an equivalence relation.
∗ +
16. LetF ={ f | f : Z → R}.For g ∈ F,wedefinetheset O(g)asfollows:
O(g) ={ f ∈ F |∃a ∈ Z ∃c ∈ R ∀x > a(| f (x)|≤ c|g(x)|)}.
+
+
(If f ∈ O(g), then mathematicians say that “ f is big-oh of g”.)
+ +
(a) Let f : Z → R and g : Z → R be defined by the formulas f (x) =
2
7x + 3 and g(x) = x . Prove that f ∈ O(g), but g /∈ O( f ).