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 2 "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 (3rded.), New York: McGraw-Hill, p.6, ISBN978-0-07-054234-1, MR0924157