Question

Let A = {a, b, c} and R = {(a, a), (a, b), (b, a)}. Then R is
(a) reflexive and symmetric but not transitive
(b) reflexive and transitive but not symmetric
(c) symmetric and transitive but not reflexive
(d) an equivalence relation

Answer 1

Hint:Here, we will use the definitions of reflexive, symmetric and transitive relations to check whether the given relations are reflexive, symmetric or transitive.

Complete step-by-step answer:
A relation between two sets is a collection of ordered pairs containing one object from each set. If the object x is from the first set and the object y is from the second set, then the objects are said to be related if the ordered pair (x, y) is in the relation.
A relation R is reflexive if each element is related itself, i.e. (a, a) $\in$ R, where a is an element of the domain.
A relation R is symmetric in case if any one element is related to any other element, then the second element is related to the first, i.e. if (x, y) $\in$ R then (y, x) $\in$ R, where x and y are the elements of domain and range respectively.
A relation R is transitive in case if any one element is related to a second and that second element is related to a third, then the first element is related to the third, i.e. if (x, y) $\in$ R and (y, z) $\in$ R then (x, z) $\in$ R.
Here, the given relation is:
R = {(a, a), (a, b), (b, a)}
The elements present in A are a, b and c.
We can see that (a, a) $\in$ R but neither of (b, b) or (c, c) exists in R.
So, R is not reflexive.
Now, we can see that (a, b) $\in$ R and (b, a) $\in$ R.
This means that R is symmetric.
Now, (a, a) $\in$ R and (a, b) $\in$ R.
For R to be transitive (a, b) must belong to R, which is true.
Therefore R is transitive.
So, R is symmetric and transitive but not reflexive.
Hence, option (c) is the correct answer.

Note: Students should note here that for a relation to be a particular type of relation, i.e. reflexive, symmetric or transitive, all its elements must satisfy the required conditions. If we are able to find even a single counter example, i.e. if any of the elements doesn’t satisfy the criteria, then we can’t proceed further.