Question

54.
RCA A (where A + 0) is an equivalence
relation if R is
[Orissa JEE 2010]
(A) Reflexive, symmetric but not transitive
(B) Reflexive, niether symmetric
transitive
(C) Reflexive, symmetric and transitive
(D) None of these
nor​

Answer 1

Answer:

Let R be a relation on A i.e. R⊆A×A

R={(a,b)∣a,b∈A}

Also, given R is equivalence relation,

Now, let R−1={(b,a)∣(a,b)∈R}

We will check whether R−1 is reflexive, symmetric, transitive or an equivalence relation.

Reflexive:

Since, R is reflexive

⇒(a,a)∈R

⇒(a,a)∈R−1  (by def of R−1)

Hence, R−1 is reflexive.

Symmetric: Let (b,a)∈R−1

⇒(a,b)∈R      (by def of R−1)

⇒(b,a)∈R  (Since, R is symmetric)

⇒(a,b)∈R−1    (by def of R−1)

Hence, R−1 is symmetric.

Transitive : Let (b,a),(a,c)∈R−1

⇒(a,b),(c,a)∈R       (by def of R−1)

or (c,a)(a,b)∈R

⇒(c,b)∈R    (since, R is transitive.)

⇒(b,c)∈R−1

Hence, R−1 is transitive.

Hence, R−1 is an equivalence relation

Step-by-step explanation:

an equialvance relation