Question

Let $A=\left\{ 1,2,3 \right\}\text{ and }R=\left\{ \left( 1,2 \right),\left( 1,1 \right),\left( 2,3 \right) \right\}$ be relation on A. Then the ordered pair to be added to R to make it the transitive relation on A.
\[\begin{align}
  & A.\left( 1,3 \right) \\
 & B.\left( 3,1 \right) \\
 & C.\left( 2,1 \right) \\
 & D.\left( 1,2 \right) \\
\end{align}\]

Answer 1

Hint:To say a relation is transitive we have to satisfy a condition that, if the relation contains (a, b) and (b, c) then it should contain (a, c). In relation R of A it contains (1, 2) and (2, 3) so we have to find the element to add.

Complete step by step answer:
In the question, we are given set A which equals to {1, 2, 3} and its relation or R is equal to {(1, 2), (1, 1), (2, 3)}. Now, we have to tell which element should be added such that the relation of A or R, such that it becomes a transitive relation on A.
We can say that a particular relation is transitive if and only if it contains elements such as (a, b) and (b, c) then it should contain (a, c) to be called as transitive.
So in the R the elements are (1,2), (1,1), (2,3) in which we will consider only (1,2), and (2,3). So, here according to the rule of transitive relation that if (a, b) and (b, c) is there then (a, c) should be there.
So, if the R or relation contains (1, 2) and (2, 3) so it should contain (1, 3).
Hence, one should consider ordered pairs (1, 3) to be added to R to be considered as a transitive relation for A.
Hence, the correct option is A.

Note: There are certain properties of transitive relation like the inverse (or converse) of a transitive relation is always transitive, the intersection of two transitive relations is transitive, the union of two transitive relations need not be transitive and lastly, the complement of a transitive relation need not be transitive.