Subset

In set theory, a subset is a set which has some (or all) of the elements of another set, called superset, but does not have any elements that the superset does not have. A subset which does not have all the elements of its superset is called a proper subset. We use the symbol ⊆ to say a set is a subset of another set. We can also use ⊂ if it is a proper subset. The symbols ⊃ ⊇ are opposite - they tell us the second element is a (proper) subset of the first.

For example:

• {1, 2, 3} is a proper subset of {-563, 1, 2, 3, 68}.

• The interval [0, 1] is a proper subset of the set of real numbers (also the set of positive numbers).

[math]\displaystyle{ [0, 1] \subset R }[/math]

[math]\displaystyle{ [0, 1] \subset R_+ }[/math]

• {46,189,1264} is its own subset, and is a proper subset of the set of natural numbers.

[math]\displaystyle{ \{ 46,189,1264\} \subseteq \{ 46,189,1264\} }[/math]

[math]\displaystyle{ \{ 46,189,1264\} \subset N }[/math]

Subset Media

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  Euler diagram showing A is a subset of B (denoted A \subseteq B) and, conversely, B is a superset of A (denoted B \supseteq A).

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  The regular polygons form a subset of the polygons.

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  A \subseteq B and B \subseteq C implies A \subseteq C.

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  A is a proper subset of B.

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  C is a subset but not a proper subset of B.

Related pages

• Subset sum problem
• Vector subspace