Three relations R1, R2 and R3 are defined on a set A = {a, b, c} as follows:
R1 = {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c, b), (c, c)}
R2 = {(a, a)}
R3 = {(b, c)}
R4 = {(a, b), (b, c), (c, a)}.
Find whether or not each of the relations R1, R2, R3, R4 on A is (i) reflexive (ii) symmetric and (iii) transitive.
Symmetric: Clearly $(a, a) \in R \Rightarrow(a, a) \in R .$ So, $\mathrm{R}_{2}$ is symmetric.
Transitive: $R_{2}$ is clearly a transitive relation, since there is only one element in it.
(i) R1
Reflexive:
Clearly, (a, a), (b, b) and (c, c)
So, R1 is reflexive.
Symmetric:
We see that the ordered pairs obtained by interchanging the components of R1 are also in R1.
So, R1 is symmetric.
Transitive:
Here,
$(a, b) \in R_{1},(b, c) \in R_{1}$ and also $(a, c) \in R_{1}$
So, R1 is transitive.
(ii) R2
Reflexive: Clearly $(a, a) \in R_{2} .$ So, $\mathrm{R}_{2}$ is reflexive.
(iii) R3
Reflexive:
Here,
$(b, b) \notin R_{3}$ neither $(c, c) \notin R_{3}$
So, R3 is not reflexive.
Symmetric:
Here,
$(b, c) \in R_{3}$, but $(c, b) \notin R_{3}$
So, $R_{3}$ is not symmetric.
Transitive:
Here, R3 has only two elements. Hence, R3 is transitive.
(iv) R4
Reflexive:
Here,
$(a, a) \notin R_{4}, \quad(b, b) \notin R_{4}(c, c) \notin R_{4}$
So, $R_{4}$ is not reflexive.
Symmetric:
Here,
$(a, b) \in R_{4}$, but $(b, a) \notin R_{4} .$
So, $R_{4}$ is not symmetric.
Transitive:
Here,
$(a, b) \in R_{4}, \quad(b, c) \in R_{4}$, but $(a, c) \notin R_{4}$
So, $\mathrm{R}_{4}$ is not transitive.