Examples for discussion 13
a. Let S = {1, 2, 3, 4) and let relation R on A be defined by
R = {(1, 2), (1, 4), (2, 3), (3, 1), (4, 2)}. Find
1) R-1
2) R′
3) R2 = RR
4) reflexive closure of R
5) symmetric closure of R
6) transitive closure of R
b. Find the properties of the following relation: R = {(a, b)N  N | a + b is even}
Reflexive: Since a + a = 2a for all natural numbers a, R is reflexive.
Symmetric: If (a, b)  R, then a + b = 2k for some integer k.
Then, b + a = 2k also, so (b, a)  R. Thus, R is symmetric.
Clearly, R is not irreflexive, antisymmetric or asymmetric, so the only thing we have left
to consider is the transitive property. Suppose (a, b)  R and (b, c)  R. Then, there
are integers j and k such that
a + b = 2k and b + c = 2j Thus,
a + c + 2b = 2k + 2j
a + c = 2(k + j - b)
Since a + c is even, (a, c)  R and R is transitive.
c. Start with A = {1, 2, 3, 4}. Find nonempty relations on A with the following characteristics:
1) irreflexive and antisymmetric
2) symmetric and transitive but not reflexive
d. Prove the complement of a symmetric relation is symmetric.
Proof: Let R be a symmetric relation on a set A and let (a, b) be an element of R′.
(a, b)  R
Definition of complement
(b, a)  R
Definition of symmetric
(b, a)  R′
Definition of complement
Therefore R′ is symmetric
Definition of symmetric
e. Prove the inverse of a transitive relation is transitive.
Proof: Let R be a transitive relation on S.
Suppose (a, b)  R-1 and (b, c)  R-1
Then, (b, a)  R and (c, b)  R
Thus, (c, a)  R
Therefore, (a, c)  R-1
Therefore R-1 is transitive
Definition of inverse
Given that R is transitive
Definition of inverse
Definition of transitive
f. One of the properties of composition is R1(R2R3)  (R1R2)  (R1R3). Find an
example for which the containment is proper.
g. Prove that a relation R on a set S is symmetric if and only if R = R-1
Part 1: If R is symmetric, then R = R-1.
In this case we must show that if R is symmetric, then R  R-1 and R-1  R.
Let (a, b)  R.
Then, (b, a)  R.
Then, (a, b)  R-1
Therefore, R  R-1.
Given that R is symmetric
Definition of inverse
Definition of subset
Let (x, y)  R-1
Then, (y, x)  R
Then, (x, y)  R.
Therefore R-1  R
Therefore R = R-1
Definition of inverse
Given that R is symmetric
Definition of subset
Definition of set equality
Part 2: If R = R-1 then R is symmetric
(Specifically, what we need to show is that if (a, b)  R, then (b, a)  R.)
Let (a, b)  R.
Then, (a, b)  R-1
Then, (b, a)  R.
Therefore, R is symmetric.
Given that R = R-1
Definition of inverse
Definition of symmetric