Solution to Let P = {1,2,3,4} and R = {(1,1), (2,1), (1,2), (2,2), (3,2), (3,4), (4,3), (4,4)} … - Sikademy
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Archangel Macsika

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.

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Answer to Question #152854 in Discrete Mathematics for prathik

Question #152854
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.
Expert's answer

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.

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Question ID: mtid-5-stid-8-sqid-3435-qpid-2134