Question

Let $A=\{1,2,3\}$ and $R=\left\{\right(2,2),(3,1),(1,3\left)\right\}$ then the relation R on A is:

• A. reflexive
• B. symmetric
• C. transitive
• D. anti-symmetric

Answer 1

The correct option is B symmetric

Given $A=\{1,2,3\}$ and $R=\left\{\right(2,2),(3,1),(1,3)$.

A relation $R$ in $A$ is said to be reflexive, if $(a,a)\in R$ for every $a\in A$.

Since $(3,3)\notin R$ and $(1,1)\notin R$, the relation $R$ on $A$ is not reflexive.

A relation $R$ in $A$ is said to be symmetric, if $(a_1,a_2)\in R⟹(a_2,a_1)\in R$ for $a_1,a_2\in A$.

$(3,1)\in R⟹(3,1)\in R$

Also $(2,2)\in R⟹(2,2)\in R$

Hence relation $R$ on $A$ is symmetric.

A relation $R$ in $A$ is said to be transitive, if $(a_1,a_2)\in R$ and $(a_2,a_3)\in R⟹(a_1,a_3)\in R$ for all $a_1,a_2,a_3\in A$.

$(3,1)\in R$ and $(1,3)\in R$ but $(3,3)\notin R$ Hence relation $R$ on $A$ is not transitive.