Proof: $a<b$ and $ b<c$, then also $a<c$ (for partially ordered set of $M$)

Question

I have the following question:

If $\leq$ is a partially ordered set over $M$ and $a<b$ and $b<c$, then also $a<c$ holds.

I'm a little confused, because I know that a partially ordered set is a transitive relation and that's exactly the statement from above. Did I miss anything and if I did, perhaps someone can help me out?

Thanks in advance.

Answer 1

I assume you define $a<b$ whenever $a\leq b\wedge a\neq b$. Then use the fact that $\leq$ is a partial order, thus it is transitive, and the additional hypothesis that $a\neq b$,$b\neq c$ to conclude.