# What does this “subset” symbol mean?

I just came across this "subset" symbol in a PDF:

$$\Omega \subsetneq T$$

I've never seen it before, and I tried looking for it via Detexify (to no avail). What does it mean?

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subset definitions –  pedja Jan 17 '12 at 9:51

This means $\Omega$ is a proper subset of $T$. That is, $\Omega\subseteq T$ but $\Omega \neq T$.

For example, $\{1, 2, 3\}\subsetneq \{1, 2, 3, 4\}$.

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@gablin: Another variant is $\subsetneqq$, and of course some folks use $\subset$ to mean proper subset. –  Brian M. Scott Jan 17 '12 at 10:23
Argh, the author mixes the various notations throughout the document! Why not just pick one and be consistent... –  gablin Jan 17 '12 at 10:28
Well, people sometimes just use $\subset$ to mean $\subseteq$. Because it is quicker to type, perhaps? Context is all-important! –  user1729 Jan 17 '12 at 10:39
@user1729 I think besides being quicker to type, it also mirrors the $<$ and $\leq$ signs nicely. –  Kris Harper Jan 17 '12 at 14:25
@root45 - that's not what I mean. Read my post again... –  user1729 Jan 17 '12 at 20:41

$\subsetneq$ ($\text{"\\subsetneq"}$) means: subset, but not equal.

Here's a nice example:

Let Ω be a half-strip in the complex plane: $$\Omega = \{ z \in \mathbb{C} | x_1 \leq \mathrm{Re} (z) \leq x_2 \text{ and } \mathrm{Im} (z) \geq y_0 \} \subsetneq \mathbb{C}. \,$$

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