Question

Let R be the relation on the set A = {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Then,
(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

Hint: For solving this problem, we consider all the necessary conditions for a set to be reflexive, symmetric and transitive. If the values in set B fail to satisfy the condition, its corresponding properties would be rejected.

Complete step-by-step answer:
The conditions which must be true for a set to be reflexive, transitive and symmetric are:
For a relation to be reflexive, $\left( a,a \right)\in R$.
For a relation to be symmetric, $\left( a,b \right)\in R\Rightarrow \left( b,a \right)\in R$.
For a relation to be transitive, $\left( a,b \right)\in R,\left( b,c \right)\in R\Rightarrow \left( a,c \right)\in R$.
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, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)} on set A = {1, 2, 3, 4}. For the set R, it contains all the elements of the form (a, a) like (1, 1) present in set A. Hence, the set R is a reflexive relation.
The set contains (1, 2) but it does not contain its counterpart, that is (2, 1), so it is not symmetric.
The set contains (1, 3), (3, 2) and (1, 2), so it is transitive by condition (3).
Therefore, the set is reflexive and transitive but not symmetric.
Hence, option (b) is correct.

Note: Students must remember all the necessary conditions for proving a set a reflexive, symmetric and transitive. All the options should be conclusively determined to give a final answer.