Question

On the set $A=\{1,2,3,4\}$, a relation is defined as $R=\left\{(1,3)(4,2)(2,4)(2,3)(3,1)\right\}$. Then $R$ is:

• A. a function
• B. transitive
• C. not symmetric
• D. reflexive

Answer 1

The correct option is C not symmetric

$A=\{1,2,3,4\}$ and $R=\left\{\right(1,2),(4,2),(2,4),(2,3),(3,1\left)\right\}$

A relation is a function if each element in its domain relates to only one element in the range.

Here, $2$ is related to $4$ and $3$. Hence $R$ is not a function.

A relation $R$ on $A$ is said to be transitive, if $(a,b)\in R$, $(b,c)\in R⟹(a,c)\in R$

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

A relation $R$ on $A$ is said to be symmetric, if $(a,b)\in R$, $⟹(b,a)\in R$

$(2,3)\in R$. But $(3,2)\notin R$. Hence the relation is not symmetric.

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

Here $(1,1),(2,2),(3,3),(4,4)$ are not elements of $R$. Hence the relation is not reflexive.