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For x,y in R, define a relation R by x R...

For `x,y in R`, define a relation R by x R y if and only if `x-y+sqrt(2)` is an irrational number. Then R is :

A

reflexive

B

symmetric

C

transitive

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
`xRy iff (x - y + sqrt(2))` is an irrational number.
Let `(x, x) in R`.
Then, `x-x+sqrt(2)=sqrt(2)` which is an irrational number,
`therefore xRx, AAx inR`
`therefore` R is an reflexive relation.
`xRyimplies(x-y+sqrt(2))` is an irrational number.
`implies-(y-x-sqrt(2))` is an irrational number.
`yRximplies(y-x+sqrt(2))` is an irrational number.
So, `xRycancelimpliesyRx thereforeR` is not a symmetric relation.
Let (1, 2) `in R`, then `(1-2+sqrt(2))` is an irrational number.
implies `(sqrt(2)-1)` is an irrational number. and (2, 3) `in R`, then `(2-3+sqrt(2))` is an irrational number.
`implies (sqrt(2)-1)` is an irrational number.
`(1, 3)inRimplies(1-3+sqrt(2))` is an irrational number.
implies `(sqrt(2)-2)` is an irrational number.
So, `(1, 2) in R and (2, 3) in R cancelimplies (1, 3) in R` (by any way)
`therefore R` is not transitive relation.
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