Solution:
Here we can find the equivalence relation on given set.
We know that
Equivalence Relation: A relation R on a set A is said to be an equivalence relation on A iff
1. It is reflexive i.e. (a, a) ∈ R for all a ∈ A.
2. It is symmetric i.e. (a, b) ∈ R
⇒ (b, a) ∈ R for all a, b ∈ A.
3. It is transitive i.e. (a , b) ∈ R and (b , c) ∈ R
⇒ (a, c) ∈ R for all a, b, c ∈ A.
We have
Let A = {p, q, r}
By definition of equivalence relation :
The given set A satisfying the reflexive, symmetric and transitive. So first we find reflexive, symmetric and transitive by using their definition.
We know that
It is reflexive (i.e.,) (a, a) ∈ R for all a ∈ A.
Here, R = { (p, p) , (q, q) , (r, r) }
We know that
It is symmetric i.e. (a, b) ∈ R
⇒ (b, a) ∈ R for all a, b ∈ A.
Here, R = (p, r) , (r, p) }
We know that
It is transitive i.e. (a , b) ∈ R and (b , c) ∈ R
⇒ (a, c) ∈ R for all a, b, c ∈ A.
Here,
R = {(p, q) (q, r) (p, r)}.
Therefore, An equivalence relation on A is R4 = { (p, p) , (q, q) , (r, r), (p, r) , (r, p)}
Hence, Let A = {p, q, r}. An equivalence relation on A is R4 = { (p, p) , (q, q) , (r, r), (p, r) , (r, p)}
Here other options are not correct because they are not satisfying the definition of equivalence relation.
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