Definition:Subset

Definition
A set $$S$$ is a subset of a set $$T$$ iff 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 $$T$$:


 * $$S \nsubseteq T$$ means $$\neg \left( {S \subseteq T}\right)$$

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:
 * $$S \subseteq T \iff \left({\forall x: x \in S \implies x \in T}\right)$$

Superset
If $$S$$ is a subset of $$T$$, then that means $$T$$ is a superset of $$S$$, which can be expressed by the notation $$T \supseteq S$$. This can be interpreted as $$T$$ contains $$S$$.

Thus $$S \subseteq T$$ and $$T \supseteq S$$ mean the same thing.

Also see

 * Compare the concept of a proper subset.

Notation in the literature can be confusing. Many authors, for example and, use $$\subset$$.

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.