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Question -

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.



Answer -

(i) Consider R1

Given R1 = {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c,b), (c, c)}

Now we have check R1 isreflexive, symmetric and transitive

Reflexive:

Given (a, a), (b, b)and (c, c) ∈ R1

So, R1 isreflexive.

Symmetric:

We see that the ordered pairs obtained by interchanging the components of R1 arealso in R1.

So, R1 is symmetric.

Transitive:

Here, (a, b) ∈R1, (b, c)∈R1 and also (a, c)∈R1

So, R1 istransitive.

(ii) Consider R2

Given R2 ={(a, a)}

Reflexive: 

Clearly (a,a) ∈R2.

So, R2 isreflexive.

Symmetric: 

Clearly (a,a) ∈R 

⇒ (a, a) ∈R.

So, R2 issymmetric.

Transitive: 

R2 isclearly a transitive relation, since there is only one element in it.

(iii) Consider R3

Given R3 = {(b, c)}

Reflexive:

Here,(b, b)∉ R3 neither (c, c) ∉ R3

So, R3 isnot reflexive.

Symmetric:

Here, (b, c) ∈R3, but (c,b) ∉R3

So, R3 isnot symmetric.

Transitive:

Here, R3 has only two elements.

Hence, R3 istransitive.

(iv) Consider R4

Given R4 = {(a, b), (b, c), (c, a)}.

Reflexive:

Here, (a, a) ∉ R4, (b, b)∉ R4 (c, c)∉ R4

So, R4 is not reflexive.

Symmetric:

Here, (a, b) ∈ R4, but (b,a) ∉ R4.

So, R4 is not symmetric

Transitive:

Here, (a, b) ∈R4, (b, c)∈R4, but (a, c)∉R4

So, R4 is not transitive.

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