# Let

Question:

Let A = {1, 2, 3}, and let R1 = {(1, 1), (1, 3), (3, 1), (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.

Solution:

(1)1 R1
Reflexivity:
Here,

$(1,1),(2,2),(3,3) \in R$

So, $R_{1}$ is reflexive.

Symmetry:

Here,

$(2,1) \in R_{1}$, but $(1,2) \notin R_{1}$

So, $R_{1}$ is not symmetric.

Transitivity:

Here, $(2,1) \in R_{1}$ and $(1,3) \in R_{1}$, but $(2,3) \notin R_{1}$

So, $R_{1}$ is not transitive.

(2)2 R2
Reflexivity:

Clearly, $(1,1)$ and $(3,3) \notin R_{2}$

So, $R_{2}$ is not reflexive.

Symmetry:

Here, $(1,3) \in R_{2}$ and $(3,1) \in R_{2}$

So, $R_{2}$ is symmetric.

Transitivity:

Here, $(1,3) \in R_{2}$ and $(3,1) \in R_{2}$

But $(3,3) \notin R_{2}$

So, $R_{2}$ is not transitive.

(3)3 R3
Reflexivity:

Clearly, $(1,1) \notin R_{3}$

So, $R_{3}$ is not reflexive.

Symmetry:

Here, $(1,3) \in R_{3}$, but $(3,1) \notin R_{3}$

So, $R_{3}$ is not symmetric.

Transitivity:

Here, $(1,3) \in R_{3}$ and $(3,3) \in R_{3}$

Also, $(1,3) \in R_{3}$

So, $R_{3}$ is transitive.

Here,