Question

Let R be a relation on set A = {1, 2, 3} defined as R = {(1, 1), (2, 2), (3, 3)}. Then, R is
(a)reflexive
(b)symmetric
(c)transitive
(d)all of the three

Answer 1

Hint: For solving this problem, we consider all options individually. By using the necessary conditions for a set to be reflexive, symmetric and transitive, we proceed for solving the question. If any of the options fails to satisfy the condition, it would be rejected.

Complete step-by-step answer:
The conditions which must be true for a set to be reflexive, transitive and symmetric are:
1)For a relation to be reflexive, $\left( a,a \right)\in R$.
2)For a relation to be symmetric, $\left( a,b \right)\in R\Rightarrow \left( b,a \right)\in R$.
3)For a relation to be transitive, $\left( a,b \right)\in R,\left( b,c \right)\in R\Rightarrow \left( a,c \right)\in R$.
4)For a relation to be equivalence, it should be reflexive, symmetric and transitive.
According to the problem statement, we are given a relation set R = {(1, 1), (2, 2), (3, 3)} on A = {1, 2, 3}. For the set R, it contains all the elements of the form (a, a) present in set A. Hence, the set R is a reflexive relation.
The set R contains any element of the form (1, 1) and (1, 1), so it is symmetric because (a, b) and (b, a) are present. The set R is transitive because it contains (1, 1) which can be related as a = b = c = 1 to satisfy the conditions of point (3). Hence, it is also not equivalence.
Therefore, option (d) is correct.

Note: This problem can be alternatively solved by proving relation R as identity relation. Since the set R contains all the elements of the form (a, a), so it is an identity relation. Identity relations are always equivalence relations.