Question

State the reason for the relation R in the set {1, 2, 3} given by R={(1, 2),(2,1)) not to be transitive.

Answer 1

Hint: In a set with elements {x,y,z} a relation R is transitive if the relation R has the elements {(a,b),(b,c), (a,c)}. So compare what is given and what is missing in order to get the reason why the given relation R={(1, 2),(2,1)) is not transitive.

Complete step-by-step answer:
In the question, we have to state the reason for the relation R in the set {1, 2, 3} given by R={(1, 2),(2,1)) not to be transitive. So here it can be recalled that if there are three elements in set and that is given as: set A={x,y,z}, then the relation R will be transitive if the elements of the set R are as follows: set R={(a,b),(b,c), (a,c)}.
So now as per the question here we have the set {1, 2, 3}, so here we have x=1, y=2 and z=3. Now, the given relation in the question is R={(1, 2),(2,1)) and when compared with x, y and z, then we have: R={(a,b),(b,c)}. So, here there is a missing element (a,c) or (1,3). So from this we can clearly say the given relation R={(1, 2),(2,1)) not to be transitive for the set elements {1, 2, 3}.

Note: It can be noted that the order matters here, so (1,2) is the order pair, and this is not equal to the (2,1) in the relation, until and unless the relation is symmetric and it should be also mentioned in the question.