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Question
Mark the correct alternative in the following question:
For real numbers x and y, define xRy if `x-y+sqrt2` is an irrational number. Then the relation R is ___________ .
Options
reflexive
symmetric
transitive
none of these
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Solution
We have,
R = {`(x, y) : x−y+sqrt2` is an irrational number; x, y ∈ R}
As, `x−x+sqrt2 = sqrt2`, which is an irrational number
⇒ (x, x) ∈ R
So, R is reflexive relation
Since, (`sqrt2`, 2) ∈ R
i.e. `sqrt2−2+sqrt2=2sqrt2−2`, which is an irrational number
but `2−sqrt2+sqrt2=2`, which is a rational number
⇒ (2, `sqrt2`) ∉ R
So, R is not symmetric relation
Also, (`sqrt2`, 2) ∈ R and (2, `2sqrt2`) ∈ R
i.e. `sqrt2−2+sqrt2=2sqrt2−2`, which is an irrational number and `2−2sqrt2+sqrt2=2−sqrt2`, which is also an irrational number
But `sqrt2−2sqrt2+sqrt2=0`, which is a rational number
⇒ `(sqrt2, 2sqrt2)` ∉ R
So, R is not transitive relation
Hence, R is Reflexive.
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