Definition:Subset

A set $$S$$ is a subset of a set $$T$$ if and only if all of the elements of $$S$$ are also elements of $$T$$, and it is written $$S \subseteq T$$.

This can also be read as "$$S$$ is contained in $$T$$", or "$$T$$ contains $$S$$".

If the elements of $$S$$ are not all also elements of $$T$$, then $$S$$ is not a subset of $$T$$:

$$S \not \subseteq T$$ means $$\lnot \left( {S \subseteq T}\right)$$

For example, if $$S = \left\{ {1, 2, 3} \right\}$$ and $$T = \left\{ {1, 2, 3, 4} \right\}$$, then $$S \subseteq T$$.

So, if we can prove that if an element is in $$S$$ then it is also in $$T$$, then we have proved that $$S$$ is a subset of $$T$$.

That is, $$S \subseteq T \iff \left(x \in S \Longrightarrow x \in T\right)$$.

Alternatively, the notation $$T \supseteq S$$ means the same thing, meaning "$$T$$ contains "$$S$$".

Note the difference between $$x \in T$$ and $$S \subseteq T$$.

We can see that "is a subset of" is a relation. Given any two sets $$S$$ and $$T$$, we can say that either $$S$$ is or is not a subset of $$T$$.

Proper Subset
If $$\left({S \subseteq T}\right) \land \left({S \ne T}\right) \land \left({S \ne \varnothing}\right)$$, then $$S$$ is referred to as a proper subset of $$T$$.

The set $$T$$ properly contains the set $$S$$.

If $$S \subseteq T$$ and $$S \ne T$$, then the notation $$S \subset T$$ and $$T \supset S$$ is used.

If we wish to refer to a set which we specifically require not to be empty, we can denote it like this:

$$\varnothing \subset S$$

... and one which we want to specify as possibly being null, we write:

$$\varnothing \subseteq S$$

Thus for $$S$$ to be a proper subset of $$T$$, we can write it as $$\varnothing \subset S \subset T$$.

Notes:

Some authors do not require that $$S \ne \varnothing$$ for $$S$$ to be a proper subset of $$T$$.

The literature can be confusing. Many authors use $$\subset$$ for what we have defined $$\subseteq$$ for. If it is important with this usage to indicate that $$S$$ is a proper subset of $$T$$, the notation $$S \subsetneq T$$ or $$T \supsetneq S$$ can be used.