Question

Check whether the relation S in the set of real numbers R defined by S = {(a, b): where a – b + `sqrt2` is an irrational number} is reflexive, symmetric or transitive.

Answer 1

For a Relation to be Reflexive aRa

For real a

aRa ⇒ a − a + `sqrt2 = sqrt2`

`sqrt2` is an irrational number 

aRa is Reflexive

For a Relation to be symmetric

aRb ⇒ bRa

For real number a and b

aRb ⇒ a − b + `sqrt2` ⇒ aRb ≠ bRa

aRb ⇒ b − a + `sqrt2` It is not symmetric

For Transitive,

aRb = bRc = aRc

For real number a, b and c

Let a = `−sqrt2`

`b = 3 sqrt2`

c = 2

aRb ⇒ `a − b + sqrt2 = −sqrt2 − 3sqrt2 + sqrt2`

= `−3sqrt2` is an irrational

bRc ⇒ `3sqrt2 − 2 + sqrt2`

= `4sqrt2 − 2` is an irrational

aRc ⇒ `−sqrt2 − 2 + sqrt2`

= −2 is not an irrational

aRb, bRc then a is not related to b.

The Relation is not transitive.