for real number x and y,define a relationR,xRy if and only if x-y+21/2is an irrational number.then the relation R is ?

Hello Student,
Please find solution to your question below [x, y \in R] Relation is defined x R y such that x – y + (2)1/2is an irrational number.
Let a to be an irrational number. Then, [x - y + \sqrt{2} = a]
[y = x + \sqrt{2}-a]
This is similar to straight line.
So for every input, there is one output So, R is one – one relation. Since range is equal to co domain, R is onto also. Finaaly, R is one – one & onto relation.

Ans:
Hello Student,
Please find answer to your question below
x R y such that
x – y + (2)^1/2is an irrational number.
Let a be an irrational number.
x – y + (2)^1/2= a
y = x + (2)^1/2– a
This is straight line.
So R is one – one relation.
Also the range is equal to co domain.
So R is one- one & onto relation.