Subset
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In set theory, a set <math>A</math> is called a subset of a set <math>B</math> if all of the elements of <math>A</math> are contained in <math>B</math>. For example, any set is a subset of itself. Another example of a subset is a proper subset: a set <math>A</math> is called a proper subset of a set <math>B</math> if <math>A</math> is subset of <math>B</math> but is not equal to <math>B</math>.
The symbol "<math>\subseteq</math>" always means "is a subset of."[1][2][3] The symbol "<math>\subsetneq</math>" always means "is a proper subset of." There is also the symbol "<math>\subset</math>", 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:
- <math>\{3,7\}</math> is a subset of <math>\{3,7\}</math>, so we could write <math>\{3,7\} \subseteq \{3,7\}</math>.
- <math>\{3,7\}</math> is a proper subset of <math>\{1,3,4,7\}</math>, so we could write <math>\{3,7\} \subseteq \{1,3,4,7\}</math>,<math>\{3,7\} \subsetneq \{1,3,4,7\}</math>, or <math>\{3,7\} \subset \{1,3,4,7\}</math>.
- The interval [0, 1] is a proper subset of the set of real numbers <math>\mathbb{R}</math>, so <math>[0, 1] \subset \mathbb{R}</math>.
Related pages[change]
References[change]
- ↑ 1.0 1.1 "Comprehensive List of Set Theory Symbols". Math Vault. 2020-04-11. Retrieved 2020-08-23.
- ↑ Weisstein, Eric W. "Subset". mathworld.wolfram.com. Retrieved 2020-08-23.
- ↑ "Introduction to Sets". www.mathsisfun.com. Retrieved 2020-08-23.
- ↑ Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, p. 6, ISBN 978-0-07-054234-1, MR 0924157