Reading Quiz Section 7.4
1. Which of the following are true and which false?
(a) [28] = [5] in Z
6
.
(b) [24] +
[3] + [17]
= [−10] in Z
9
.
(c) [2]
3
+ [3]
3
= [4]
3
in Z
29
.
2. Is the following True or False?
In Z
n
, [x] + [y] = [z] ⇐⇒ x + y = z
3. Let ∼ be an equivalence relation on X. What does it mean for a function f :
X
∼
→ B to be
well-defined?
(a) f is injective.
(b) [x] = [y] if and only if f ([x]) = f ([ y]).
(c) f is surjective.
(d) x ∼ y implies f ([x]) = f ([y]).
4. True or False: the rule [x] 7→ x : Z
n
→ Z is well-defined.
Practice Problems Section 7.4
1. Working in Z
n
, use B
´
ezout’s identity to prove that
∃b such that [a] · [b] = [1]
⇐⇒ gcd(a, n) = 1
2. Let k be a constant integer. If f
[x]
5
= [kx]
18
is a well-defined function f : Z
5
→ Z
18
, what
can you say about the sequence of values
f ([0]
5
), f ([1]
5
), f ([2]
5
), . . . ?
That is, when must they start repeating themselves?
3. In Exercise 7.4.13 show that F is surjective if and only if f is a bijection.
