Question

Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.
(a)R is reflexive and symmetric but not transitive
(b) R is reflexive and transitive but not symmetric
(c)R is symmetric and transitive but not reflexive
(d)R is 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 = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}.
Let us call the domain set A.
So, the elements present in A are 1, 2, 3 and 4.
In the relation R, we can see that for any element a $\in $ A, (a, a) always exists in R.
So, R is reflexive.
Now, take any ordered pair (a, b) $\in $ R, we see that (b, a) does not exist in R.
For example, if we take (1, 2) $\in $ R, we see that (2, 1) $\notin $ R.
This means that R is not symmetric.
Now, take any ordered pair (a, b) $\in $ R and (b, c) $\in $ R.
We can see that (a, c) always belongs to R.
For example, if we take (1, 2) $\in $ R and (2, 2) $\in $ R then we can see also (1, 2) $\in $ R.
Again, (1, 3) $\in $ R and (3, 3) $\in $ R, then (1, 3) $\in $ R
This implies that R is transitive.
Therefore, R is reflexive and transitive but not symmetric.
Hence, option (b) 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.