Question

Let \[A = \left\{ {0,1,2,3} \right\}\] and define a relation R as follows \[R = \left\{ {\left( {0,0} \right),\left( {0,1} \right),\left( {0,3} \right),\left( {1,0} \right),\left( {1,1} \right),\left( {2,2} \right),\left( {3,0} \right),\left( {3,3} \right)} \right\}\]. Is R reflexive, symmetric and transitive?

Answer 1

Hint: In this question, we are given a set A and we are given a relation in this set. We have to find whether the relation is reflexive, symmetric and transitive. So, all the conditions for a set to be reflexive, symmetric and transitive are considered. If the relation fails to satisfy those conditions, then it wouldn’t qualify as reflexive, symmetric or transitive.

Complete step-by-step solution:
We have \[A = \left\{ {0,1,2,3} \right\}\] and \[R = \left\{ {\left( {0,0} \right),\left( {0,1} \right),\left( {0,3} \right),\left( {1,0} \right),\left( {1,1} \right),\left( {2,2} \right),\left( {3,0} \right),\left( {3,3} \right)} \right\}\]
The necessary condition for a relation to be reflexive is $(a,a) \in R$
For set A, all the elements of the form (a,a) are present in R. For example, $(0,0) \in R$
So, the given relation is reflexive.
For a set to be symmetric, the necessary condition is that if $(a,b) \in R$ then $(b,a) \in R$
The set R contains both $(0,1)$ and $(1,0)$ . It also contains $(3,0)$ and $(0,3)$ so the given relation is a reflexive relation.
The necessary condition for a set to be transitive is that if the set contains $(a,b)$ and $(b,c)$ then it must contain $(a,c)$ .
The set R contains $(1,0)\,and\,(0,3)$ but it doesn’t contain $(1,3)$ . So, the given relation is not transitive.
Hence, the given relation is symmetric and reflexive but not transitive.

Note: The set must pass all the conditions that are necessary for it to become reflexive, symmetric or transitive. Some sets can pass one of the conditions but may not pass the others. The relations that are reflexive, symmetric and transitive are known as equivalent relations. The given relation is not an equivalence relation.