Let `A={p,q,r}`. Which of the following is an equivalence relation on `A`?
(i) `R={(p,q),(q,r),(p,r),(p,p)}`
(ii) `R={(p,p),(q,q),(r,r),(q,p)}`
(iii) `R={(p,p),(q,q),(r,r)}`
(iv) `R={(p,p),(q,q),(r,r),(p,q),(q,r),(p,r)}`
(v) `R={(p,p),(q,q),(r,r),(p,q),(q,p)}`

Options

A. `R_(1) = {(p, q), (q, r), (p, r), (p, p)}`
B. `R_(2) = {(r, q), (r, p), (r, r), (q, q)}`
C. `R_(3) = {(p, p), (q, q), (r, r), (p, q)}`
D. None of the above

Answer

D

Explanation

A = {p, q, r}
`R_(1) = {(p, q), (q, r), (p, r), (p, p)}`
`(q, q) cancelin R_(1)`, so `R_(1)` is not reflexive relation
So, `R_(1)` is not an equivalence relation.
`R_(2) = {(r, q), (r, p), (r, r), (q, q)}`
Here, `(p, p) cancelin R_(2)`, so `R_(2)` is not an equivalence relation .
`R_(3)={(p, p),(q, q), (r, r), (p, q)}`
`R_(3)` is an reflexive relation.
`(p, a) in R_(3) " but "(q, p) cancelinR_(3)`
`R_(3)` is not symmetric relation.
So, `R_(3)` is not equivalence relation.