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If Z is the set of integers. Then, the r...

If Z is the set of integers. Then, the relation `R={(a,b):1+ab gt 0}` on Z is

A

reflexive and transitive but not symmetric

B

symmetric and transitive but not reflexive

C

reflexive and symmetric but not transitive

D

an equivalence relation

Text Solution

Verified by Experts

The correct Answer is:
C
Given relation is `R={(a,b):1+ab gt 0}`
Reflexive A relation `R sube A xx "A is reflexive, if (a, a)" in R`.
Now, `" "R={(a,b):1+ab gt0}" on Z."`
`implies" "(a, a)in Rimplies1+a^(2)gt0`
So, it is reflexive.
Symmetric if `(a, b) in R implies (b, a) in R`
Since, `1+ab gt0implies1+ba gt 0`
`implies" "(b,a)in R`
So, it is symmetric.
Transitive if `(a,b) in R" and "(b,c)in Rimplies(a,c) in R`
Now, `" "(a,b)in R" and "(b,c)inR`
`implies" "1+ab gt0" and "1+bcgt0`
`implies" "2+b(a+c)gt0`
i.e., `(a,c) !in R`.
So, it is not transitive.
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