Is relation R transitive or not?

Question

R = { (1, 1), (2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4) }

I know that this relation is transitive BUT I just want to make sure that this relation is really transitive. Having elements of ordered pairs (1, 1) and (2, 2) and (3, 3), do these elements affect transitivity? Is a relation a transitive relation if there are elements that satisfy the condition and there are other elements that do not satisfy the condition?

Answer 1

The only way to really be sure that the relation is transitive is to test each pair of pairs to see if the result is in the relation.

For example (2,3) (3,2) -> (2,2)

Having (1,1) and (2,2) and (3,3) in the relation makes this easier - you don't have to bother considering them in any of the pairings to prove transitivity since they are "identity operators" - including them in any pairing simply returns the other pair, intrinsically consistent with transitivity.

That leaves
(2,3) (2,4) (3,2) (3,4)

There aren't that many combinations to consider, just brute force it.

(2,3) (3,2) -> (2,2) ok
(2,3) (3,4) -> (2,4) ok
(3,2) (2,3) -> (3,3) ok

Answer 2

Try each possibility of pairs of ordered pairs. For each set of two ordered pairs, you should be able to find a third that is the transitive result.
A relation is only transitive if it is always transitive. You can't say that 1,2,3,4,3,8,10,17 are in order except for the problem of 3 following 4. The numbers are in order or they aren't.