CS 112
Discrete Mathematics for CS
Problem Set-3
Relations:
1. Let R1 and R2 be symmetric relations. Is R1 ∩ R2 also symmetric? Is R1 ∪ R2
also symmetric?
2. Which of these are equivalence relations on the set of all people?
a) {(x, y) | x and y were born in the same year}
b) {(x, y) | x and y have lived in the same city}
3. Show that the relation {(a, b) | a ≡ ±b (mod 7)} is an equivalence relation on the
set of integers.
4. Given the following digraph representation of a relation R:
a. What is the binary matrix representation of R?
b. Find the matrix and digraph representation of R2 = R R
5. Which of these relations on the set of all people are equivalence relations?
Determine the properties of an equivalence relation that the others lack.
a. {(a, b) | a and b are the same age}
b. {(a, b) | a and b have the same parents}
c. {(a, b) | a and b share a common parent}
d. {(a, b) | a and b have met}
e. {(a, b) | a and b speak a common language}
6. Draw the directed graph that represents the relation
R = {(a, a), (a, b), (b, c), (c, b), (c, d), (d, a), (d, b)}.
7. List the ordered pairs in the relations on {1, 2, 3} corresponding to these matrices
(where the rows and columns correspond to the integers listed in increasing
order).
8. Let R1 = {(1, 2), (2, 3), (3, 4)} and R2 = {(1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (3, 1),
(3, 2), (3, 3), (3, 4)} be relations from {1, 2, 3} to {1, 2, 3, 4}. Find
a) R1 ∪ R2.
b) R1 ∩ R2.
c) R1 − R2.
d) R2 − R1.
9. Let R be the relation {(1, 2), (1, 3), (2, 3), (2, 4), (3, 1)}, and let S be the relation
{(2, 1), (3, 1), (3, 2), (4, 2)}. Find S◦R.