Let R be a relation on Z x Z defined by (a, b) R (c, d) if and only if ad - bc is divisible by 5.

Question

Let \(\mathrm{R}\) be a relation on \(\mathrm{Z} \times \mathrm{Z}\) defined by \((a, b) R(c, d)\) if and only if \(a d-b c\) is divisible by 5.

Then \(\mathrm{R}\) is

(1) Reflexive and symmetric but not transitive

(2) Reflexive but neither symmetric not transitive

(3) Reflexive, symmetric and transitive

(4) Reflexive and transitive but not symmetric

Answer 1

Correct option is (1) Reflexive and symmetric but not transitive

\((\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{a}, \mathrm{b})\) as \(\mathrm{ab}-\mathrm{ab}=0\)

Therefore reflexive

Let \((a, b) R(c, d) \Rightarrow a d-b c\) is divisible by 5

\(\Rightarrow \mathrm{bc}-\mathrm{ad}\) is divisible by \(5 \Rightarrow(\mathrm{c}, \mathrm{d}) \mathrm{R}(\mathrm{a}, \mathrm{b})\)

Therefore symmetric

Relation not transitive as \((3,1) \mathrm{R}(10,5)\) and \((10,5) \mathrm{R}(1,1)\) but \((3,1)\) is not related to \((1,1)\).