Question

`R_(1)` on Z defined by `(a,b)inR_(1) " iff "|a-b|le7, R_(2)` on Q defined by `(a,b)inR_(2) " iff "ab=4 and R_(3)` on R defined by `(a, b)inR_(3)" iff "a^(2)-4ab+3ab^(2)=0`
Relation `R_(2)` is
A. reflexive
B. symmetric
C. transitive
D. equivalence

Answer 1

Correct Answer - B
We have (a, b) `in R_(2)` iff ab = 4, where a, b `in` Q
Reflexivity `5 in Q` and (5)(5) = `25 ne 4`
`therefore (5, 5) cancelin R_(2)`
The relation `R_(2)` is not reflexive.
Symmetry
(a, b) `in R_(2)`
`implies ab = 4 implies ba = 4`
`implies (b, a) in R_(2)`
`therefore` The relation `R_(2)` is symmetric.
Transitivity We have `(8,(1)/(2)),((1)/(2),8)inR_(2)` because
`8((1)/(2))=4 and ((1)/(2))(8)=4`
Also, 8(8) = 64 `ne` 4
`therefore (8, 8) cancelin R_(2)`
`therefore` The relation `R_(2)` is not transitive.