Question

Let $P=\{1,2,3\}$ and a relation on set $P$ is given by the set $R=\left\{\right(1,2),(1,3),(2,1),(1,1),(2,2),(3,3),(2,3\left)\right\}$. Then $R$ is:

• A. Reflexive, transitive but not symmetric
• B. Symmetric, transitive but not reflective
• C. Symmetric, reflexive but not transitive
• D. None of the above

Answer 1

The correct option is A Reflexive, transitive but not symmetric

GIven $P=\{1,2,3\}$ and $R=\left\{\right(1,2),(1,3),(2,1),(1,1),(2,2),(3,3),(2,3\left)\right\}$

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

Since $(1,1),(2,2),(3,3)\in R$ , $R$ is reflexive.

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 $a_1,a_2,a_3\in A$.

$(1,2)\in R$ , $(2,3)\in R$ and $(3,3)\in R$

$(2,1)\in R$ , $(1,3)\in R$ and $(2,3)\in R$

Hence relation $R$ on $P$ is transitive.

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

$(1,3)\in R$ But $(3,1)\notin R$

Hence $R$ is not symmetric.