Question

Let R = {(a, a), (b, b), (c, c), (a, b)} be a relation on set A = {a, b, c}. Then, R is
(a)identity relation
(b)reflexive
(c)symmetric
(d)equivalence

Answer 1

Hint: For solving this problem, we consider all options individually. By using the necessary conditions for a set to be identity, 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:
For a relation to be identity, it should contain all $\left( a,a \right)\in R$ of the parent set.
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 = {(a, a), (b, b), (c, c), (a, b)} on A = {a, b, c}. For the set R, it contains all the elements of the form (a, a), (b, b) and (c, c) present in set A. Hence, the set R is identity relation as well as reflexive relation by using the above conditions.
The set R contains an element (a, b) but it does not contain (b, a), so it cannot be symmetric. Since the set is not symmetric, it is also not equivalent.
Therefore, option (a) and (b) are correct.

Note: The knowledge of equivalence of a relation is must for solving this problem. 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.