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My book says that $\subset$ is used to represente any subset, proper or improper, needing in this case to show the anti symmetric property of sets. ($A = B \iff A \subset B \, \, \wedge \,\, B \subset A)$

And $\subseteq$ is used specifically to represent improper subsets. (In other words, $A \subseteq B \iff A = B) $

But i saw so many articles using $\subseteq$ and not $\subset$ then i am kinda suspicious about this definition.

If this definition is correct, why $\subseteq$ it's so used? Just because it helps to prove the equality betwen the sets?

Thanks for the help.

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marked as duplicate by Daniel Fischer, egreg, Lord_Farin, T. Bongers, Bruno Joyal Nov 26 '13 at 13:25

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Your suspicion is justified. I advice you to use $\subset$ for subsets (proper as well as improper), and $\subsetneq$ for subsets that are proper when that is a relevant thing in the context. – drhab Nov 26 '13 at 11:49
What book are you reading? I have never seen $\subseteq$ used to denote equality between sets. – arjafi Nov 26 '13 at 11:55
See the first answer to this question:… – universalset Nov 26 '13 at 11:57
I had a professor (a logician by practice) who was a bit picky about notation. He never used $\subset$ and always used $\subseteq$ in its place. He also avoided $a/b$ when writing $\frac{a}{b}$, and threatened to give me a $0$ on a quiz because my fraction bar didn't fully cover the denominator. He was a cool guy though. – doppz Nov 26 '13 at 12:49
@ArthurFischer ok,so in what situation i may use $\subseteq$? – Voyager Nov 26 '13 at 21:48
up vote 4 down vote accepted

It is common practice to use $\subset$ to simply denote a subset, proper or not. $\subseteq$ is used to mean the same thing. This is mainly due to reasons of convenience. I'm a pedant and prefer to use $\subset$ for proper subset, whereas $\subsetneq$ is convention for proper subsets.

I disagree that $\subseteq$ is used to represent improper subsets since we already have a good symbol for that: $=$

In reading though, you should interpret $\subset$ as subset, proper or not. Context typically makes it clear though.

(Now imagine if primary and secondary educators decided to go on protest and use $<$ instead of $\leq$ for reasons of convenience).

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ok, so in what situation i may use $\subseteq$? – Voyager Nov 26 '13 at 21:49
I simply use it for subset. Basically one has three choices of the (subset,proper subset) combination: $(\subseteq,\subset)$, $(\subseteq,\subsetneq)$, and $(\subset,\subsetneq)$ with the last one being most common, although I prefer the first. The key thing is to just make sure your audience knows which one you're using. – Andrew Nov 26 '13 at 22:26

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