Sets And Relations Question 5
Question 5 - 2024 (29 Jan Shift 1)
Let $\mathrm{R}$ be a relation on $\mathrm{Z} \times \mathrm{Z}$ defined by $(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{c}, \mathrm{d})$ if and only if $\mathrm{ad}-\mathrm{bc}$ 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
Show Answer
Answer (1)
Solution
$(a, b) R(a, b)$ as $a b-a b=0$
Therefore reflexive
Let $(a, b) R(c, d) \Rightarrow a d-b c$ is divisible by $5 \Rightarrow \mathrm{bc}-$ 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 $(1,1)$
