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Not sure how to answer this question.

"Show that inclusion is an order relation in the set of all sets"

My understanding is that I have to show that inclusion on the set of all sets, is reflexive, anti-symmetric and transitive. Is this right? I'm not sure why the set of all sets is included in the question, but it must be there for a reason?

What is the difference between "order relation" and "partial order"?

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  • $\begingroup$ I think there may be more than one type of def'n of partial order. In the set-theoretic topic of Forcing, a poset (short for partially ordered set) is a set with a binary relation $ \leq$ that is reflexive and transitive. It is allowable in Forcing to have a poset in which $x\leq y\leq x$ and $x\ne y$ for some $x,y.$ $\endgroup$ – DanielWainfleet Mar 26 '17 at 18:11
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"The set of all sets"?

Gosh.

Is your lecturer a closet NF-iste?

Seriously though, a partial order on the sets (or if you want, on the proper class of sets) must be what is meant -- as plainly inclusion is not a total order (why not?).

And the difference between "order relation" and "partial order"? We might say that, on the one hand, that there are a variety of different order relations, mere partial orderings being only one kind. On the other hand, perhaps every order worthy of the label is at least a partial ordering (so that so-called pre-order aren't, in this sense, order relations).

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