Question

Let $R = \left\{ {\left( {a,a} \right)} \right\}$ be a relation on set A. then R is
1) Symmetric
2) Antisymmetric
3) Symmetric and antisymmetric
4) Neither symmetric nor antisymmetric

Answer 1

Hint: This question is based on the concept of relations. There are various types of relations like reflexive, symmetric, transitive, and equivalence relations. These must be known to the student to solve the problem.
A relation is an equivalence if and only if it is Reflexive, Symmetric, and Transitive. For example, if we throw two dice A & B, note down all the possible outcomes.

Complete step-by-step answer:
So, the relation given to us in the problem is: $R = \left\{ {\left( {a,a} \right)} \right\}$.
Now, we know that if every element of set A maps to itself, then the relation is known as a reflexive relation.
Now, we are given that $\left( {a,a} \right)$ belongs to the relation R. So, we have, \[aRa\]. Therefore, $\left( {a,a} \right) \Leftrightarrow \left( {a,a} \right)$ is true. Therefore, the given relation is reflexive.
Similarly, a relation R on a set A is said to be symmetric if both the elements $\left( {a,b} \right)$ and $\left( {b,a} \right)$ belong to the relation. Now, there is only one pair $\left( {a,a} \right)$ in the relation R. Since $\left( {a,a} \right)$ is the symmetric pair of itself and we know that $\left( {a,a} \right)$ belongs to the relation R. So, the relation R is symmetric.
Antisymmetric relation is a relation in which $\left( {a,b} \right)$ belongs to relation R only when $a = b$.
Since, we know that there is only one element in the given relation and that $\left( {a,a} \right)$ belongs to R. So, the given function is antisymmetric as $\left( {a,a} \right) \Leftrightarrow \left( {a,a} \right)$.
Therefore, the given relation is both symmetric and antisymmetric.
So, the correct answer is “Option 3”.

Note: In mathematics, a set is a collection of elements. The set with no element is the empty set; a set with a single element is a singleton; otherwise, a set may have a finite number of elements or be an infinite set.