Question

Let we have a set as $A=\left\{ 1,2,3,4 \right\}$ and $R=\left\{ \left( 2,2 \right),\left( 3,3 \right),\left( 4,4 \right),\left( 1,2 \right) \right\}$ be a relation on A. Find the characteristics of R.
A. reflexive
B. symmetric
C. transitive
D. none of these

Answer 1

Hint: We first try to find the definition of different characteristics like reflexive, symmetric, transitive. The given relation R is on A. We take components from R to find if the terms follow the rules of the characteristics. The relation only follows the transitive property.

Complete step-by-step solution
We have been given a set $A=\left\{ 1,2,3,4 \right\}$ and a relation R on A where $R=\left\{ \left( 2,2 \right),\left( 3,3 \right),\left( 4,4 \right),\left( 1,2 \right) \right\}$.
We need to find the characteristics of R.
For R to be reflexive $\left( a,a \right)\in R$ is must $\forall a\in A$.
Now $1\in A$ but $\left( 1,1 \right)\notin R$. So, R is not reflexive.
Now for R to be symmetric $\left( b,a \right)\in R$ is must if $\left( a,b \right)\in R$, $\forall a,b\in A$.
Now $\left( 1,2 \right)\in R$ but $\left( 2,1 \right)\notin R$. So, R is not symmetric.
For R to be transitive $\left( a,c \right)\in R$ is must if $\left( a,b \right)\in R$ and $\left( b,c \right)\in R$, $\forall a,b,c\in A$.
Now $\left( 1,2 \right)\in R$ and $\left( 2,2 \right)\in R$ which also gives $\left( 1,2 \right)\in R$. So, R is transitive. We don’t need to simplify for the reflexive terms.
The correct option is C.

Note: We need to remember that the relation for A is from A to $A\times A$. We are taking certain duals from the set of $A\times A$. The number of terms in $A\times A$ is ${{n}^{2}}$ when the number of terms in A is n.