Question

Let A = $\left\{ 1,2,3 \right\}$ and R = $\left\{ \left( 1,2 \right),\left( 1,1 \right),\left( 2,3 \right) \right\}$ be a relation on A. What minimum number of ordered pairs must be added to R so that it may become a transitive relation on A.

Answer 1

Hint: First we are going to look at the definition of symmetric, reflexive and transitive. And after that we will look at what ordered pair needs to be added so that it can become a transitive relation on A.

Complete step-by-step answer:
Let’s start our solution by first writing all the definition of the terms:
Symmetric: If we have a set containing two elements ‘a’ and ‘b’, then if the relation set has (a,b) then it must have (b,a) then we can say that it is symmetric.
Reflexive: If we have a set containing two elements ‘a’ and ‘b’, then if the relation set has (a,a) and (b,b) then we can say it is reflexive.
Transitive: If we have a set containing three elements ‘a’ , ‘b’, and ‘c’ then if the relation set has (a,b) and (b,c) then it must have (a,c) for transitive.
Now we have stated all the required definitions.
As we can see that R= $\left\{ \left( 1,2 \right),\left( 1,1 \right),\left( 2,3 \right) \right\}$ has (1,2) and (2,3) so for it to be transitive it must have (1,3).
Hence we just need to add one ordered pair which is (1,3).

Note: All the definitions of the terms that we have used is very important, without understanding it’s meaning clearly we cannot solve this question. And with that we need to find the minimum number of ordered pairs which is also very important.