Question

A relation S in the set of real numbers is defined as `"xSy" => "x" - "y" + sqrt3`  is an irrational number, then relation S is ____________.

Options

• reflexive

• reflexive and symmetric

• transitive

• symmetric and transitive

Answer 1

A relation S in the set of real numbers is defined as `"x S y" => "x" - "y" + sqrt3`  is an irrational number, then relation S is reflexive.

Explanation:

reflexive, true as `"x S x" => "x" - "x" + sqrt 3 = sqrt3` is an irrational number.

Symmetric, false e.g. `"x" = sqrt3 , "y" = 2`

`"xSy" => sqrt3 - 2 + sqrt3 = 2 sqrt3 - 2` is an irrational number.

but `"ySx" => 2 - sqrt3 + sqrt3  2` is not an irrational number.

transitive, false e.g. `"x" = 1 + sqrt3, "y" = 5, "z" = 2 sqrt3`

`"xSy" => 1 + sqrt3 - 5 + sqrt3 = 2sqrt3 - 4` is an irrational number.

`"ySz" => 5 - 2sqrt3 + sqrt3 = 5 - sqrt3` is an irrational number.

But `"xSz" => 1 + sqrt3 - 2sqrt3 + sqrt3 = 1` not an irrational number.