Question

Let $R=\left\{ \left( a,a \right) \right\}$ be a relation on a set A. Then R is
(A). Symmetric
(B). Anti-Symmetric
(C). Transitive
(D). Neither symmetric nor antisymmetric

Answer 1

Hint: In this question, we are given the relation in terms of an ordered pair and we have to find out the properties it satisfies. Therefore, we need to understand the definitions of the symmetric, anti-symmetric, transitive properties and check if the given relation satisfies those definitions to obtain the answer to this question.

Complete step-by-step solution -
We are given that the relation R is defined as $R=\left\{ \left( a,a \right) \right\}$. Therefore, it associates only the element a of the set A to itself.
The definition of symmetric, anti-symmetric and transitive properties for a relation between the sets A and B are as follows
1). A relation R from A to B is said to be symmetric when for every ordered pair \[\left( a,b \right)\] belonging R, the pair \[\left( b,a \right)\] also belongs to R……..(1.1)
2). A relation R from A to B is said to be anti-symmetric when for every ordered pair \[\left( a,b \right)\] belonging R, the pair \[\left( a,b \right)\] is also belongs to R only if a=b……………..(1.2)
3). A relation R from A to B is said to be transitive when for every ordered pair \[\left( a,b \right)\] and \[\left( b,c \right)\] belonging to R, the pair \[\left( a,c \right)\] also belongs to R……………………..(1.3)
Now, we see that in the given relation $R=\left\{ \left( a,a \right) \right\}$, the only ordered pair is \[\left( a,a \right)\]. Therefore, exchanging the elements a and a, we find the ordered pair still is \[\left( a,a \right)\] which is in R. Therefore, using equation (1.1) the relation is symmetric.
Also, we see that as there is only one ordered pair \[\left( a,a \right)\], and obviously $a=a$, by using equation (1.2), we find that the relation R is also anti-symmetric.
Now, as a is only related to a by the relation R, then, the pairs \[\left( a,b \right)\], \[\left( b,c \right)\] and \[\left( a,c \right)\] are all equal to \[\left( a,a \right)\] are all the same and therefore the relation R is also transitive.
Thus, we find that the relation R is symmetric, anti-symmetric and transitive.

Note: We should note that in the equations (1.1), (1.2) and (1.3), there is no restriction that b or c cannot be a. Different notations are chosen just to show that the relation should hold if the other variables b or c are equal or not equal to a.