Let R be a relation on N × N defined by a , b R c , d if and only if a d b - c = b c a - d . Then R is

Let R be a relation on N×N defined by a,bRc,d  if and only if adb-c=bca-d. Then R is
  1. symmetric but neither reflexive nor transitive
  2. transitive but neither reflexive nor symmetric
  3. reflexive and symmetric but not transitive
  4. symmetric and transitive but not reflexive

Solution

Given, a,bRc,d⇒adb-c=bca-d

⇒a,bRc,d⇔b-cbc=a-dad

⇒1c-1b=1d-1a

⇒1a-1b=1d-1c

Reflexive:

a,bRa,b⇒1a-1b=1b-1a

It is false. So, given relation is not reflexive.

Symmetric:

a,bRc,d⇒1a-1b=1d-1c

⇒1c-1d=1b-1a

∴c,dRa,b

It is symmetric.

Transitive:

a,bRc,d⇒1a-1b=1d-1c

c,dRe,f=1c-1d=1e-1f

1a-1b=1e-1f

⇒a,bRe,f

So, not transitive.

Asked in: JEE Main 2023 (31 Jan Shift 1)

Practice more Sets and Relations questions on Aicharya