Subset

In set theory, a set [math]\displaystyle{ A }[/math] is called a subset of a set [math]\displaystyle{ B }[/math] if all of the elements of [math]\displaystyle{ A }[/math] are contained in [math]\displaystyle{ B }[/math]. For example, any set is a subset of itself. Another example of a subset is a proper subset: a set [math]\displaystyle{ A }[/math] is called a proper subset of a set [math]\displaystyle{ B }[/math] if [math]\displaystyle{ A }[/math] is subset of [math]\displaystyle{ B }[/math] but is not equal to [math]\displaystyle{ B }[/math].

A is a subset of B.

The symbol "[math]\displaystyle{ \subseteq }[/math]" always means "is a subset of." The symbol "[math]\displaystyle{ \subsetneq }[/math]" always means "is a proper subset of." There is also the symbol "[math]\displaystyle{ \subset }[/math]", which some authors use to mean "is a subset of" and other authors only use to mean "is a proper subset of."

For example:

• [math]\displaystyle{ \{3,7\} }[/math] is a subset of [math]\displaystyle{ \{3,7\} }[/math], so we could write [math]\displaystyle{ \{3,7\} \subseteq \{3,7\} }[/math].

• [math]\displaystyle{ \{3,7\} }[/math] is a proper subset of [math]\displaystyle{ \{1,3,4,7\} }[/math], so we could write [math]\displaystyle{ \{3,7\} \subseteq \{1,3,4,7\} }[/math],[math]\displaystyle{ \{3,7\} \subsetneq \{1,3,4,7\} }[/math], or [math]\displaystyle{ \{3,7\} \subset \{1,3,4,7\} }[/math].

• The interval [0, 1] is a proper subset of the set of real numbers [math]\displaystyle{ \mathbb{R} }[/math], so [math]\displaystyle{ [0, 1] \subset \mathbb{R} }[/math].

Subset Media

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

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  The regular polygons form a subset of the polygons.

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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