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R(1) on Z defined by (a,b)inR(1) " iff "...

`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 and symmetric

B

symmetric and transitive

C

reflexive and transitive

D

equivalence

Text Solution

Verified by Experts

The correct Answer is:
A
We have, (a, b) `in R_(1)` iff |a-b| `le 7`, where `a, b in z`
Reflexivity Let `a in z`
implies a - a = 0
`implies |a - a| le 7`
`implies 0 le 7`
`implies (a, a) in R_(1)`
`therefore` The relation `R_(1)` is reflexive.
Symmetry
`(a, b) in R_(1)`
`implies |a-b|le7implies|-(b-a)|le7`
`implies |b-a|le7implies(b, a)in R_(1)`
`therefore` The relation `R_(1)` is symmetric.
Transitivity We have (2, 6), (6, 10) `in R_(1)` because
`|2-6|=4le7 and |6-10|=4le7`
Also, |2-10| = `8cancelle7`
`therefore (2, 10) cancelinR_(1)`
Hence, the relation `R_(1)` is not transitive.
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