Discrete Math: How is this relation property not transitive

I don't understand the answer to this question.

Given the set {1, 2, 3, 4, 5}

Determine what properties this relation has

R = { (1,2), (2,3), (3,4), (4,5) }

It is irreflexive, and antisymmetric

It is NOT reflexive, symmetric and transitive.

I understand all of them but how is it not transitive?

When evaluating these problems R can be anything (such as =, <, etc)

If it was < then wouldn't it be transitive? Because x < y AND y < z thus x < z ?

Or am I wrong about how this works?

A relation R on a set S is said to be transitive if for all a, b, and c in S, whenever (a,b) and (b,c) are in R, so is (a,c). In this example, (1,2) and (2,3) are in R, but (1,3) is not, so R is not transitive.

(1,2) is the same as 1 R 2, i.e. 1 and 2 are related. We have (1,2) and (2,3), but not (1,3), so this is not transitive.