How to determine transitivity and intransitivity of this relation?

Question

I am having trouble finding if the following relation is transitive or intransitive. I would be very thankful if someone could help me out by explaining transitivity and its rules with regard to this example,

$$R = \{ (1,3),(1,1),(3,1),(1,2),(3,3)(4,4) \} .$$

Thanks.

Answer 1

A relation on a set is transitive if, when we have (a,b) and (b,c), we have also (a,c).

So here, we just check some cases. 4 is on on its own - so we don't really care about 4.

Let's look at 1 and 3. We have (1,3) and (3,1) - do we have (1,1)? Yes, we do. Similarly, we have (3,1) and (1,3), and we also have (3,3). These are all the relations on 3.

But then we consider 1 and 2...

Answer 2

Transitivity says that if you have $aRb$ and $bRc$ you also have $aRc$. When your relation is a given set of pairs, you have to look through and see if it is true. If $a=2$, there is no $b$ such that $aRb$, so we satisfy the requirement. If $a=4$, we must have $b=4$ and then $c=4$ and we do get $aRc$. When $a$ is $1$ or $3$, you have more possibilities for $b$ and have to follow each path to see where it leads.