On proper use of notations $\in$, $\subset$, $\subseteq$

Suppose a number of objects form set $M$. Furthermore, pairs of objects from $M$ form another set $W$. I'm interested in proper mathematical use of the following notations:

• when referring to a particular member of $W$, one writes $\{a, b\}\in W$, or $(a, b)\in W$ (of course, I ask about the convention), $\{a, b\}\subset W$, or $(a, b)\subseteq W$ (?)

• when one wants to state that object $a$ belongs to $M$, then $a\in M$ (?)

• when one wants to state that objects $a, b$ and $c$ belong to $M$, then $\{a, b, c\}\in M$, or $\{a, b, c\}\subset M$, or $\{a, b, c\}\subseteq M$. Given the above assumption, one would write (for some object pair $a, b$) $(a, b)\in W$, $\{a, b\}\subseteq V$ (?)

The above complicates with the notion of ordered/unordered pairs. How would the above notation be under such different statements?

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If you just want to say that $a$, $b$ and $c$ belong to $M$, then $\{a,b,c\}\subseteq M$ is correct but clunky. $\{a,b,c\}\in M$ is not right. You'll often just see "$a$,$b$,$c \in M$". –  mjqxxxx Jan 3 '13 at 15:04

So let $M$ be a set and $W=\left\{(a,b):a,b\in M\right\}$. $W$ is by the way called the Cartesian Product $M\times M=M^2$.

If $x\in W$ then $x=(a,b)$ for some $a,b\in M$. We thus write $(a,b)\in W$ or if you prefer $\left\{(a,b)\right\}\subseteq W$. If in addition $(a,b)$ is not the only element of $W$ then we can also write $\left\{(a,b)\right\}\subset W$.

To state $a$ belongs to $M$ one writes $a\in M$ or $\left\{a\right\}\subseteq M$.

If $a,b,c\in M$ then you can equivalently write $\left\{a,b,c\right\}\subseteq M$. One should write $(a,b)\in W$.

All these follow easily if you understand the definition of $W$ and what the notation actually stands for.

Note: Here $\subset$ is used for proper inclusion (as $\subsetneq$).

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One strong caveat for using $\subset$, there are places using it for improper inclusion and there are other places using it for proper inclusion only. I much much prefer $\subseteq$ and $\subsetneq$ (or even $\subsetneqq$). –  Asaf Karagila Jan 3 '13 at 15:09
@AsafKaragila Yes there are many writers who do that (Rudin comes to mind), though to me it isn't a big deal (as in most cases one doesn't care about whether or not the inclusion is proper). The notation you use is unambiguous and preferable however. –  Nameless Jan 3 '13 at 15:14
Thanks! One more thing: when referring to all objects, one shall write $\forall a\in M$ and $\forall (a,b)\in W$? –  user506901 Jan 3 '13 at 15:18
@user506901 All objects of what? If it's all the objects of $M$ then one writes $\forall x\in M$. If it's all the objects of $W$ then one writes $\forall x\in W$ or if you prefer $\forall (a,b)\in W$. –  Nameless Jan 3 '13 at 15:21