Definition:Subset

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[edit] 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)


[edit] 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.


[edit] Also see

Notation in the literature can be confusing. Many authors, for example A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968) and Allan Clark: Elements of Abstract Algebra (1971), 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.


[edit] Notes

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.


[edit] Sources

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