Mathematical Reasoning- Writing and Proof, Version 2.1, 2014a

384 Chapter 7. Equivalence Relations
7. Repeat Exercise (6) using the function f W R ! R that is defined by f.x/ D
x 2 3x 7 for each x 2 R.
8. (a) Repeat Exercise (6a) using the function f W R ! R that is defined by
f.x/ D sin x for each x 2 R.
(b) Determine all real numbers in the set C Dfx 2 R j x g.
9. Define the relation on Q as follows: For a; b 2 Q, a b if and only
if a b 2 Z. In Progress Check 7.9, we showed that the relation is an
equivalence relation on Q.
(a) List four different elements of the set C D x 2 Q j x 5
.
7
(b) Use set builder notation (without using the symbol ) to specify the set
C .
(c) Use the roster method to specify the set C .
10. Let and be relations on Z defined as follows:
For a; b 2 Z, a b if and only if 2 divides a C b.
For a; b 2 Z, a b if and only if 3 divides a C b.
(a) Is an equivalence relation on Z? If not, is this relation reflexive,
symmetric, or transitive?
(b) Is an equivalence relation on Z? If not, is this relation reflexive,
symmetric, or transitive?
11. Let U be a finite, nonempty set and let P.U / be the power set of U .That
is, P.U / is the set of all subsets of U . Define the relation on P.U / as
follows: For A; B 2 P.U /, A B if and only if A \ B D;.Thatis,the
ordered pair .A; B/ is in the relation if and only if A and B are disjoint.
Is the relation an equivalence relation on P.U /? If not, is it reflexive,
symmetric, or transitive? Justify all conclusions.
12. Let U be a nonempty set and let P.U / be the power set of U .Thatis,P .U /
is the set of all subsets of U .
For A and B in P.U /, defineA B to mean that there exists a bijection
f W A ! B. Prove that is an equivalence relation on P.U /.
Hint: Use results from Sections 6.4 and 6.5.