Assignment:
Q1. 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 x 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?
Q2. Let A = {3, 4, 5} and B = {4, 5, 6} and let S be the "divides" relation. That is, for all (x, y) ∈ A X 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.
Q3. R = {(0, 0), (0, 1), (1, 1), (1, 2), (2, 2), (2, 3)}.
Q4. R = {(0, 0), (0, 1), (0, 2), (1, 2)}.
Q5. R7 = {(0, 3), (2, 3)}.
Q6. 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).
Q7. 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?
Q8. Determine which of the following congruence relations are true and which are false:
a) 4 ≡ -5 (mod 7).
b) –6 ≡ 22 (mod 2).
Q9. 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).
Q10. Give a real-world example of a relation which is (and justify why it is): symmetric.
Provide complete and step by step solution for the question and show calculations and use formulas.