Question

Let A = {2, 4, 6, 8}. A relation R on A is defined by R = {(2,4),(4,2),(4,6),(6,4)}
Then the R is-
A. Anti-symmetric
B. Reflexive
C. Symmetric
D. Transitive

Answer 1

Hint: Any relation can be classified as reflexive, symmetric and transitive. If aRa exists in the relation, then it is said to be reflexive. If aRb and bRa both exist in the relation, then it is said to be symmetric. If aRb and bRc exist implies that aRc also exists, the relation is transitive. Using these definitions we will check the relation R.

Complete step-by-step answer:
Before starting with the solution, let us discuss different types of relations. There are a total of 8 types of relations that we study, out of which the major ones are reflexive, symmetric, transitive, and equivalence relation.

Reflexive relations are those in which each and every element is mapped to itself, i.e., $\left( a,a \right)\in R$ . Symmetric relations are those for which, if $\left( a,b \right)\in R\text{ }$ then $\left( b,a \right)$ must also belong to R. This can be represented as $aRb\Rightarrow bRa$ . Now, transitive relations are those for which, if $\left( a,b \right)\text{ and }\left( b,c \right)\in R$ then $\left( a,c \right)$ must also belong to R, i.e., $\left( a,b \right)\text{ and }\left( b,c \right)\in R\Rightarrow \left( a,c \right)\in R$ .
Now, if there exists a relation, which is reflexive, symmetric, and transitive at the same time, then the relation is said to be an equivalence relation. For example: let us consider a set A=(1,2). Then the relation {(1,2),(2,1),(1,1),(2,2)} is an equivalence relation.

Now let us start with the solution to the above question.We have been given that-
R = {(2,4),(4,2),(4,6),(6,4)} on the set A = {2, 4, 6, 8}.
For R to be reflexive, (a, a) should be an element of R for all values of a belongs to A. But this is not true. (2, 2), (4, 4) and so on are not the elements of R. So, R is not reflexive.

For R to be symmetric, if (a, b) is an element of R then (b, a) is also an element of R. We can clearly see that both (2, 4) and (4, 2) are elements of R. Similarly, (4, 6) and (6,4) are elements of R. Hence, R is a symmetric relation.

For R to be transitive, if (a, b), (b, c) are elements of R, then (a, c) is also the element of R. We can see that (2, 4) and (4, 6) are elements of R but the same cannot be said for (2, 6). Hence, R is not transitive.
R is only a symmetric relation. Hence, the correct option is C.

Note: It is important to check carefully for each condition. It is also recommended to check and verify each condition using a suitable example. Even if one case is false, the condition is not verified. Also if it is not possible to prove that relation is symmetric, reflexive or transitive, then use a suitable example to show that it is not.