Question

Let the relation an a set $A=\left\{ 0,1,2,3 \right\}$ is define as $R=\left\{ \left( 0,0 \right),\left( 0,1 \right),\left( 0,3 \right),\left( 1,0 \right)\left( 1,1 \right),\left( 2,2 \right)\left( 3,0 \right),\left( 3,3 \right) \right\}$. Is R reflexive, symmetric and transitive?

Answer 1

Hint: To solve this problem, we should know the properties of reflexive relation, symmetric relation and transitive relation. We know that for a relation R defined on the set $A=\left\{ a,b,c,d... \right\}$ to be reflexive, R should contain all the elements such as $\left( a,a \right),\left( b,b \right)....\forall a,b,c...\in A$. We can write that the relation R is symmetric, if an element $\left( a,b \right)\in R$ then $\left( b,a \right)$ should be present in R. We can write that the relation R is transitive, if $\left( a,b \right),\left( b,c \right)\in R$ then $\left( a,c \right)$ should be present in the set R. Using these three properties, we can verify whether the given relation is reflexive, symmetric and transitive.

Complete step-by-step answer:
We are given a relation $R=\left\{ \left( 0,0 \right),\left( 0,1 \right),\left( 0,3 \right),\left( 1,0 \right)\left( 1,1 \right),\left( 2,2 \right)\left( 3,0 \right),\left( 3,3 \right) \right\}$ on a set $A=\left\{ 0,1,2,3 \right\}$. We are asked to find the properties of the relation.
We know that for a relation R defined on the set $A=\left\{ a,b,c,d... \right\}$ to be reflexive, R should contain all the elements such as $\left( a,a \right),\left( b,b \right)....\forall a,b,c...\in A$.
Let us consider the reflexive property of the relation. From the above rule, we have the values of $a=0,b=1,c=2,d=3$ which are elements of the set A. For the relation R to be reflexive, every element of $\left( 0,0 \right),\left( 1,1 \right),\left( 2,2 \right),\left( 3,3 \right)$ should be present in the set R.
We can observe that the elements $\left( 0,0 \right),\left( 1,1 \right),\left( 2,2 \right),\left( 3,3 \right)\in R$, so, we can conclude that the relation R is reflexive.
We can write that the relation R is symmetric, for every element $\left( a,b \right)\in R$ we should have an element $\left( b,a \right)$ in R.
Let us consider the symmetric property in the given relation$R=\left\{ \left( 0,0 \right),\left( 0,1 \right),\left( 0,3 \right),\left( 1,0 \right)\left( 1,1 \right),\left( 2,2 \right)\left( 3,0 \right),\left( 3,3 \right) \right\}$
We have $\left( 0,1 \right)$ in the relation R and the symmetric element $\left( 1,0 \right)$ is also present in R.
We have $\left( 0,3 \right)$ in the relation R and the symmetric element $\left( 3,0 \right)$ is also present in R.
So, for every element $\left( a,b \right)\in R$, we have the symmetric element $\left( b,a \right)$ in R. So, we can conclude that T is symmetric.
We can write that the relation R is transitive, if $\left( a,b \right),\left( b,c \right)\in R$ then $\left( a,c \right)$ should be present in the set R. Let us consider the transitive property in the relation R.
We have $\left( 1,0 \right)\text{ and }\left( 0,3 \right)$ in the relation R but we don’t have $\left( 1,3 \right)$ in the relation R. SO, we can conclude that R is not transitive.
$\therefore $ The given relation R is reflexive and symmetric but not transitive in nature.

Note: We should note the fact that the reflexive property is only property which has a condition that every element in the form $\left( a,a \right)$ should be present in the relation. The other two properties come into picture if there is an element of the form $\left( a,b \right)$. The symmetric and transitive properties depend on the elements present in the relation but the reflexive property has a separate type of condition that every element should be present in the relation.