Question

Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer. (A) R is reflexive and symmetric but not transitive. (B) R is reflexive and transitive but not symmetric. (C) R is symmetric and transitive but not reflexive. (D) R is an equivalence relation.

Answer 1

The given relation R in the set { 1,2,3,4 } is defined as R={( 1,2 ),( 2,2 ),( 1,1 ),( 4,4 ),( 1,3 ),( 3,3 ),( 3,2 )}.

( 1,1 ),( 2,2 ),( 3,3 ),( 4,4 )∈R.

So R is reflexive.

( 1,2 )∈R but ( 2,1 )∉R.

So, Ris not symmetric.

Let, a, b and c be elements in the given set such that ( a,b )and ( b,c )∈R.

It is observed that ( a,c )∈Rfor all ( a,b,c )∈{ 1,2,3,4 }. So Ris transitive.

The given relation, R is reflexive and transitive but not symmetric.

Hence, option (B) is correct.