Question

Let P = {1,2,3,4} and R = {(1,1), (2,1), (1,2), (2,2), (3,2), (3,4), (4,3), (4,4)} which is a relation on P. Represent this relation as a directed graph. Check whether this relation is an equivalence relation or not.

Answer 1

We draw the directed graph with the vertex set $P=$ {1,2,3,4} and an edge from $i$ to $j$ whenever $(i,j)$ is in the given relation

$R=$ {$(1,1),(2,1),(1,2),(2,2),(3,2),(3,4),(4,3),(4,4)$ }.

The directed graph representing the relation can be used to determine whether the relation is reflexive, symmetric or transitive. A relation is reflexive if and only if there is a loop at every vertex of the directed graph, so that every ordered pair is of the form $(i,i)$ occurs in the relation. But here in the above directed graph we see that there is no loop at the vertex $3$.

So the given relation $R$ is not reflexive.

Again we know that a relation $R$ is said to be an equivalence relation on $P$ iff $R$ is reflexive, symmetric and transitive relation. But here the given relation $R$ is not reflexive relation.

Therefore $R$ is not an equivalence relation.