Three relations `R_1, R_2 and R_3` are defined on set `A={a , b , c}` as follow:
`R_1={(a , a),(a , b),(a , c),(b , b),(b , c),(c , a),(c , b),(c , c)}`
`R_2={(a , a)}`
`R_3={(b,c)}`
`R_4={(a , b),(b , c),(c , a)}`
Find whether each of `R_1, R_2,R_3,R_4` is
`(i) ` reflexive,
`(ii) `symmetric and
`(iii) `transitive.

Explanation

`R_1 = {(a,a),(a,b),(a,c),(b,b),(b,c),(c,a),(c,b),(c,c)}`

As `R_1` contains `(a,a),(b,b) and (c,c)`, so, it is reflexive.

As `R_1` contains `(a,b)` but do not contain `(b,a)`, so it is not symmetric.

As `R_1` contains `(a,b),(b,c)` and `(a,c)`, so it is transitive.



`R_2 = {(a,a)}`


As `R_2` contains `(a,a)`,but, do not contain `(b,b) and (c,c)`, so, it is not reflexive.

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