Let A = {1, 2, 3}, and let R1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3), (2, 2), (2, 1), (3, 3)}, R2={(2,2),(3,1), (1, 3)}, R3 = {(1, 3),(3, 3)}. Find whether or not each of the relations R1, R2, R3 on A is (i) reflexive (ii) symmetric (iii) transitive.
We have been given,
A = {1, 2, 3}
Here, R1, R2, and R3 are the binary relations on A.
So, recall that for any binary relation R on set A. We have,
R is reflexive if for all x ∈ A, xRx.
R is symmetric if for all x, y ∈ A, if xRy, then yRx.
R is transitive if for all x, y, z ∈ A, if xRy and yRz, then xRz.
So, using these results let us start determining given relations.
Let us take R1.
R1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}
(i). Reflexive:
∀ 1, 2, 3 ∈ A [∵ A = {1, 2, 3}]
(1, 1) ∈ R1
(2, 2) ∈ R2
(3, 3) ∈ R3
So, for a ∈ A, (a, a) ∈ R1
∴ R1 is reflexive.
(ii). Symmetric:
∀ 1, 2, 3 ∈ A
If (1, 3) ∈ R1, then (3, 1) ∈ R1
[∵ R1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}]
But if (2, 1) ∈ R1, then (1, 2) ∉ R1
[∵ R1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}]
So, if (a, b) ∈ R1, then (b, a) ∉ R1
∀ a, b ∈ A
∴ R1 is not symmetric.
(iii). Transitivity:
∀ 1, 2, 3 ∈ A
If (1, 3) ∈ R1 and (3, 3) ∈ R1
Then, (1, 3) ∈ R1
[∵ R1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}]
But, if (2, 1) ∈ R1 and (1, 3) ∈ R1
Then, (2, 3) ∉ R1
So, if (a, b) ∈ R1 and (b, c) ∈ R1, then (a, c) ∉ R1
∀ a, b, c ∈ A
∴ R1 is not transitive.
Now, take R2.
R2 = {(2, 2), (3, 1), (1, 3)}
(i). Reflexive:
∀ 1, 2, 3 ∈ A [∵ A = {1, 2, 3}]
(1, 1) ∉ R2
(2, 2) ∈ R2
(3, 3) ∉ R2
So, for a ∈ A, (a, a) ∉ R2
∴ R2 is not reflexive.
(ii). Symmetric:
∀ 1, 2, 3 ∈ A
If (1, 3) ∈ R2, then (3, 1) ∈ R2
[∵ R2 = {(2, 2), (3, 1), (1, 3)}]
If (2, 2) ∈ R2, then (2, 2) ∈ R2
[∵ R2 = {(2, 2), (3, 1), (1, 3)}]
So, if (a, b) ∈ R2, then (b, a) ∈ R2
∀ a, b ∈ A
∴ R2 is symmetric.
(iii). Transitivity:
∀ 1, 2, 3 ∈ A
If (1, 3) ∈ R2 and (3, 1) ∈ R2
Then, (1, 1) ∉ R2
[∵ R2 = {(2, 2), (3, 1), (1, 3)}]
So, if (a, b) ∈ R2 and (b, c) ∈ R2, then (a, c) ∉ R2
∀ a, b, c ∈ A
∴ R2 is not transitive.
Now take R3.
R3 = {(1, 3), (3, 3)}
(i). Reflexive:
∀ 1, 3 ∈ A [∵ A = {1, 2, 3}]
(1, 1) ∉ R3
(3, 3) ∈ R3
So, for a ∈ A, (a, a) ∉ R3
∴ R3 is not reflexive.
(ii). Symmetric:
∀ 1, 3 ∈ A
If (1, 3) ∈ R3, then (3, 1) ∉ R3
[∵ R3 = {(1, 3), (3, 3)}]
So, if (a, b) ∈ R3, then (b, a) ∉ R3
∀ a, b ∈ A
∴ R3 is not symmetric.
(iii). Transitivity:
∀ 1, 3 ∈ A
If (1, 3) ∈ R3 and (3, 3) ∈ R3
Then, (1, 3) ∈ R3
[∵ R3 = {(1, 3), (3, 3)}]
So, if (a, b) ∈ R3 and (b, c) ∈ R3, then (a, c) ∈ R3
∀ a, b, c ∈ A
∴ R3 is transitive.