Title: FUNCTIONS ASSIGNMENT
Author: user
Last saved by: Devin Burns
Company: host

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.	R_7 = {(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