Question

A relation R on \[A = \{ 1,2,3\} \] defined by \[R = \{ \left( {1,1} \right)\left( {1,2} \right)\left( {3,3} \right)\} \] is not symmetric. Why?

Answer 1

Hint: We have been given set \[A = \{ 1,2,3\} \] and there exists a relation on it that is \[R = \{ \left( {1,1} \right)\left( {1,2} \right)\left( {3,3} \right)\} \]. Here, first we need to determine what is a symmetric relation and then we will apply the condition for a symmetric relation on R. It will become clear to us which parameter or condition R is not fulfilling such that it is not symmetric which will give us the answer.

Complete step by step answer:
We have A, the set in which relation R is defined.
Now let us first consider the condition for a symmetric relation.
We know that a relation R is said to be symmetric, if \[\left( {a,b} \right) \in R\] then \[\left( {b,a} \right) \in R\].
Now we will apply this condition to the elements of relation R given to us in the question.
We have, \[R = \{ \left( {1,1} \right)\left( {1,2} \right)\left( {3,3} \right)\} \]
Let us check one by one.
For (1,1) it is clear that it belongs to R, which satisfies the condition.
For (1,2), we have (1,2) belonging to R, but (2,1) does not belong to R. Hence the condition is not satisfied.
For (3,3), it is again clear that it belongs to R, satisfying the condition for symmetric relation.
Now, we know that (2,1) does not belong to the given relation R, and hence relation R is not symmetric.

Note: We have to remember that for a relation R to be symmetric, if (a,b) belongs to R then (b,a) should also belong to R. This condition has to be satisfied for all the elements in the relation R. If the condition is not satisfied for one element then the relation R is said to be not symmetric. We also have two other relations- reflexive and transitive.
Reflexive relation: A relation is said to be a reflexive relation on a given set, if each element of the set is related onto itself. For example, let A be a set and R is a relation defined on A. Then R is said to be reflexive if for each\[a \in A\,and\,\left( {a,a} \right) \in R\].
Transitive Relation: Let A be any set. A relation R on A is said to be a transitive relation if\[\left( {a,b} \right) \in R\,and\,\left( {b,c} \right) \in R\,then\,\left( {a,c} \right) \in R\,for\,all\,a,b,c \in A\].
If a relation R on set A is reflexive, symmetric and transitive then it is called an equivalence relation, provided set A is non empty.