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

The symbol "${\displaystyle \subseteq }$" always means "is a subset of."^[1][2][3] The symbol "${\displaystyle \subsetneq }$" always means "is a proper subset of." There is also the symbol "${\displaystyle \subset }$", which some authors use to mean "is a subset of"^[4] and other authors only use to mean "is a proper subset of."^[1]

For example:

• ${\displaystyle \{3,7\}}$ is a subset of ${\displaystyle \{3,7\}}$, so we could write ${\displaystyle \{3,7\}\subseteq \{3,7\}}$.

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

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

• Subset sum problem
• Vector subspace

1. ↑ ^{1.0 1.1} "Comprehensive List of Set Theory Symbols". Math Vault. 2020-04-11. Retrieved 2020-08-23.
2. ↑ Weisstein, Eric W. "Subset". mathworld.wolfram.com. Retrieved 2020-08-23.
3. ↑ "Introduction to Sets". www.mathsisfun.com. Retrieved 2020-08-23.
4. ↑ Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, p. 6, ISBN 978-0-07-054234-1, MR 0924157

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