Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I found the following statement in a book after the usual definition of preorder and preordered set and I am not sure about how I should interpret it.

"While a preordered set $(X, \succsim)$ is not a set, it is convenient to talk as if it is a set when referring to properties that apply only to $X$. For instance, by a 'finite preordered set', we understand a preordered set $(X, \succsim)$ with $|X|< \infty$."

So the question (probably trivial) is, in the end is the preordered set a set or not?

My interpretation is that here the only set is $X$, and even if we put a preorder on it, and we call the result a preordered set, still the only set around is $X$, even if we had some characteristics (i.e. the predorder) that describe the behavior of the elements of $X$ with more details.

Is this the right interpretation of this paragraph?

PS I was not sure about the title of this question. I hope it is ok.

share|cite|improve this question
up vote 2 down vote accepted

This is a small, but sometimes relevant, key issue. Are ordered pairs sets?

Recall that preordered sets are ordered pairs, $(X,R)$ such that $X$ is a [non-empty] set and $R$ is a preorder of $X$. If you treat ordered pairs as non-set objects, then preorders are not sets for that very reason.

But when we talk about the preordered set $(X,R)$, we are saying that $X$ is a set, and it is preordered by $R$. This is how you should interpret this paragraph. Often, however, it is very good to confuse between $(X,R)$ and $X$. It helps us read things better, as the example about finiteness proposes.

share|cite|improve this answer
Thanks a lot for the clarification. – Kolmin May 30 '13 at 19:31

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.