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 1

(i) R₁
Reflexive:
Clearly, (a, a), (b, b) and (c, c) ∈ R₁

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 not symmetric.

Transitive:
Here,

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

So, R₁ is transitive.

(ii) R₂

Reflexive: Clearly (a,a) ∈R₂ . So, R₂ is reflexive.

Symmetric: Clearly (a,a) ∈R ⇒ (a,a) ∈R. So, R₂ is symmetric.

Transitive: R₂ is clearly a transitive relation, since there is only one element in it.

iii) R₃
Reflexive:
Here,

(b, b)∉ R₃ neither (c, c) ∉ R₃

So, R₃  is not reflexive.

Symmetric:
Here,

(b, c) ∈R₃, but (c,b) ∉R₃

So,R₃isnotsymmetric.

Transitive:
Here, R₃ has only two elements. Hence, R₃ is transitive.

(iv) R₄
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.