# Set Relations Binary Relations

Relations

1. Let C = {2, 3, 4, 5} and D = {3, 4} and define a binary relation S from C to D as follows:

for all (x, y) C D, (x, y) S x y.

a) Write S as a set of ordered pairs.

b) Is 2 S 4? Is 4 S 3? Is (4, 4) S? Is (3, 2) S?

2. Let A = {3, 4, 5} and B = {4, 5, 6} and let S be the "divides" relation. That is,

for all (x, y) A B, x S y x | y.

State explicitly which ordered pairs are in S and S -1.

In the following 3 exercises binary relations are defined on the set A = {0, 1, 2, 3}. For each relation:

a) determine whether the relation is reflexive;

b) determine whether the relation is symmetric;

c) determine whether the relation is transitive.

Give a counterexample in each case in which the relation does not satisfy one of the proprieties, or justify why the property holds true.

3. R = {(0, 0), (0, 1), (1, 1), (1, 2), (2, 2), (2, 3)}.

4. R = {(0, 0), (0, 1), (0, 2), (1, 2)}.

5. R7 = {(0, 3), (2, 3)}.

6. Determine whether or not the following relation is reflexive, symmetric, transitive, or none of these. Justify your answer.

F is the congruence modulo 5 relation on Z: for all

m, n Z, m F n 5 | (m - n).

7. Let A be the set with eight elements.

a) How many binary relations on A are reflexive?

b) How many binary relations on A are both reflexive and symmetric?

8. Determine which of the following congruence relations are true and which are false:

a) 4 -5 (mod 7).

b) -6 22 (mod 2).

9. Describe the distinct equivalence classes of the following equivalence relation.

F is the relation defined on Z as follows:

for all m, n Z, m F n 4 | (m - n).

10. Give a real-world example of a relation which is (and justify why it is): symmetric

https://brainmass.com/math/recurrence-relation/set-relations-binary-relations-622012

#### Solution Preview

Relations

1. Let C = {2, 3, 4, 5} and D = {3, 4} and define a binary relation S from C to D as follows:

for all (x, y) C D, (x, y) S x y.

a) Write S as a set of ordered pairs.

b) Is 2 S 4? Is 4 S 3? Is (4, 4) S? Is (3, 2) S?

2. Let A = {3, 4, 5} and B = {4, 5, 6} and let S be the "divides" relation. That is,

for all (x, y) A B, x S y x | y.

State explicitly which ordered pairs are in ...

#### Solution Summary

This posting explains some questions on binary relations(reflexive, symmetric, transitive) and equivalence classes.