In mathematics, a set *A* is a **subset** of a set *B* if all elements of *A* are also elements of *B*; *B* is then a **superset** of *A*. It is possible for *A* and *B* to be equal; if they are unequal, then *A* is a **proper subset** of *B*. The relationship of one set being a subset of another is called **inclusion** (or sometimes **containment**). *A* is a subset of *B* may also be expressed as *B* includes (or contains) *A* or *A* is included (or contained) in *B*.

- Definitions
- Properties
- ⊂ and ⊃ symbols
- Examples of subsets
- Other properties of inclusion
- See also
- References
- Bibliography
- External links

The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation.

If *A* and *B* are sets and every element of *A* is also an element of *B*, then:

*A*is a**subset**of*B*, denoted by or equivalently*B*is a**superset**of*A*, denoted by^{ [1] }

If *A* is a subset of *B*, but *A* is not equal to *B* (i.e. there exists at least one element of B which is not an element of *A*), then:

*A*is a**proper**(or**strict**)**subset**of*B*, denoted by (or^{ [1] }^{[ circular reporting? ]}^{ [2] }^{[ better source needed ]}). Or equivalently,*B*is a**proper**(or**strict**)**superset**of*A*, denoted by (or^{ [1] }^{[ circular reporting? ]}).- The empty set, written { } or ∅, is a subset of any set
*X*and a proper subset of any set except itself.

For any set *S*, the inclusion relation ⊆ is a partial order on the set (the power set of *S*—the set of all subsets of *S*^{ [3] }) defined by . We may also partially order by reverse set inclusion by defining

When quantified, *A* ⊆ *B* is represented as ∀*x*(*x* ∈ *A* → *x* ∈ *B*).^{ [4] }

We can prove the statement *A* ⊆ *B* by applying a proof technique known as the element argument^{ [5] }:

Let sets

AandBbe given. To prove thatA⊆B,

supposethatais a particular but arbitrarily chosen element ofA,showthatais an element ofB.

The validity of this technique can be seen as a consequence of Universal generalization: the technique shows *c* ∈ *A* → *c* ∈ *B* for an arbitrarily chosen element *c*. Universal generalisation then implies ∀*x*(*x* ∈ *A* → *x* ∈ *B*), which is equivalent to *A* ⊆ *B*, as stated above.

- A set
*A*is a**subset**of*B*if and only if their intersection is equal to A.

- Formally:

- A set
*A*is a**subset**of*B*if and only if their union is equal to B.

- Formally:

- A
**finite**set*A*is a**subset**of*B*, if and only if the cardinality of their intersection is equal to the cardinality of A.

- Formally:

Some authors use the symbols ⊂ and ⊃ to indicate *subset* and *superset* respectively; that is, with the same meaning and instead of the symbols, ⊆ and ⊇.^{ [6] } For example, for these authors, it is true of every set *A* that *A* ⊂ *A*.

Other authors prefer to use the symbols ⊂ and ⊃ to indicate *proper* (also called strict) subset and *proper* superset respectively; that is, with the same meaning and instead of the symbols, ⊊ and ⊋.^{ [7] }^{ [1] } This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if *x* ≤ *y*, then *x* may or may not equal *y*, but if *x* < *y*, then *x* definitely does not equal *y*, and *is* less than *y*. Similarly, using the convention that ⊂ is proper subset, if *A* ⊆ *B*, then *A* may or may not equal *B*, but if *A* ⊂ *B*, then *A* definitely does not equal *B*.

- The set A = {1, 2} is a proper subset of B = {1, 2, 3}, thus both expressions A ⊆ B and A ⊊ B are true.
- The set D = {1, 2, 3} is a subset (but
*not*a proper subset) of E = {1, 2, 3}, thus D ⊆ E is true, and D ⊊ E is not true (false). - Any set is a subset of itself, but not a proper subset. (X ⊆ X is true, and X ⊊ X is false for any set X.)
- The set {
*x*:*x*is a prime number greater than 10} is a proper subset of {*x*:*x*is an odd number greater than 10} - The set of natural numbers is a proper subset of the set of rational numbers; likewise, the set of points in a line segment is a proper subset of the set of points in a line. These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one's initial intuition.
- The set of rational numbers is a proper subset of the set of real numbers. In this example, both sets are infinite, but the latter set has a larger cardinality (or
*power*) than the former set.

Another example in an Euler diagram:

- A is a proper subset of B
- C is a subset but not a proper subset of B

Inclusion is the canonical partial order, in the sense that every partially ordered set (*X*, ) is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example: if each ordinal *n* is identified with the set [*n*] of all ordinals less than or equal to *n*, then *a* ≤ *b* if and only if [*a*] ⊆ [*b*].

For the power set of a set *S*, the inclusion partial order is—up to an order isomorphism—the Cartesian product of *k* = |*S*| (the cardinality of *S*) copies of the partial order on {0,1} for which 0 < 1. This can be illustrated by enumerating *S* = {*s*_{1}, *s*_{2}, ..., *s*_{k}}, and associating with each subset *T* ⊆ *S* (i.e., each element of 2^{S}) the *k*-tuple from {0,1}^{k}, of which the *i*th coordinate is 1 if and only if *s*_{i} is a member of *T*.

In mathematics, the **cardinality** of a set is a measure of the "number of elements" of the set. For example, the set contains 3 elements, and therefore has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between the different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its **size**, when no confusion with other notions of size is possible.

In mathematics, especially order theory, a **partially ordered set** formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word *partial* in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable.

In the mathematical field of set theory, an **ultrafilter** on a given partially ordered set (poset) *P* is a certain subset of *P,* namely a maximal filter on *P*, that is, a proper filter on *P* that cannot be enlarged to a bigger proper filter on *P*.

A **mathematical symbol** is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.

In set theory, **Zermelo–Fraenkel set theory**, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated **ZFC**, where C stands for "choice", and **ZF** refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.

In axiomatic set theory and the branches of mathematics and philosophy that use it, the **axiom of infinity** is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.

In mathematics, in set theory, the **constructible universe**, denoted by `L`, is a particular class of sets that can be described entirely in terms of simpler sets. `L` is the union of the **constructible hierarchy**`L`_{α} . It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this, he proved that the constructible universe is an inner model of ZF set theory, and also that the axiom of choice and the generalized continuum hypothesis are true in the constructible universe. This shows that both propositions are consistent with the basic axioms of set theory, if ZF itself is consistent. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result.

In set theory, a branch of mathematics, a set *A* is called **transitive** if either of the following equivalent conditions hold:

In mathematics, an **upper set** of a partially ordered set is a subset *S* ⊆ *X* with the following property: if *s* is in *S* and if *x* in *X* is larger than *s*, then *x* is in *S*. In words, this means that any *x* element of *X* that is ≥ to some element of *S* is necessarily also an element of *S*. The term **lower set** is defined similarly as being a subset *S* of *X* with the property that any element *x* of *X* that is ≤ to some element of *S* is necessarily also an element of *S*.

This article examines the implementation of mathematical concepts in set theory. The implementation of a number of basic mathematical concepts is carried out in parallel in ZFC and in NFU, the version of Quine's New Foundations shown to be consistent by R. B. Jensen in 1969.

In model theory and related areas of mathematics, a **type** is an object that describes how a element or finite collection of elements in a mathematical structure might behave. More precisely, it is a set of first-order formulas in a language *L* with free variables *x*_{1}, *x*_{2},…, *x*_{n} that are true of a sequence of elements of an *L*-structure . Depending on the context, types can be **complete** or **partial** and they may use a fixed set of constants, *A*, from the structure . The question of which types represent actual elements of leads to the ideas of saturated models and **omitting types**.

In universal algebra and in model theory, a **structure** consists of a set along with a collection of finitary operations and relations that are defined on it.

In the mathematical field of set theory, an **ideal** is a partially ordered collection of sets that are considered to be "small" or "negligible". Every subset of an element of the ideal must also be in the ideal, and the union of any two elements of the ideal must also be in the ideal.

In mathematics, a **cardinal function** is a function that returns cardinal numbers.

**Pocket set theory** (**PST**) is an alternative set theory in which there are only two infinite cardinal numbers, ℵ_{0} and *c*. The theory was first suggested by Rudy Rucker in his *Infinity and the Mind*. The details set out in this entry are due to the American mathematician Randall M. Holmes.

In mathematics, specifically set theory, the **Cartesian product** of two sets *A* and *B*, denoted *A* × *B*, is the set of all ordered pairs (*a*, *b*) where *a* is in *A* and *b* is in *B*. In terms of set-builder notation, that is

In set theory, an **ordinal number**, or **ordinal**, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another.

This is a **glossary of set theory**.

**Finitist set theory (FST)** is a collection theory designed for modeling finite nested structures of individuals and a variety of transitive and antitransitive chains of relations between individuals. Unlike classical set theories such as ZFC and KPU, FST is not intended to function as a foundation for mathematics, but only as a tool in ontological modeling. FST functions as the logical foundation of the classical layer-cake interpretation, and manages to incorporate a large portion of the functionality of discrete mereology.

- 1 2 3 4 "Comprehensive List of Set Theory Symbols".
*Math Vault*. 2020-04-11. Retrieved 2020-08-23. - ↑ "Introduction to Sets".
*www.mathsisfun.com*. Retrieved 2020-08-23. - ↑ Weisstein, Eric W. "Subset".
*mathworld.wolfram.com*. Retrieved 2020-08-23. - ↑ Rosen, Kenneth H. (2012).
*Discrete Mathematics and Its Applications*(7th ed.). New York: McGraw-Hill. p. 119. ISBN 978-0-07-338309-5. - ↑ Epp, Susanna S. (2011).
*Discrete Mathematics with Applications*(Fourth ed.). p. 337. ISBN 978-0-495-39132-6. - ↑ Rudin, Walter (1987),
*Real and complex analysis*(3rd ed.), New York: McGraw-Hill, p. 6, ISBN 978-0-07-054234-1, MR 0924157 - ↑
*Subsets and Proper Subsets*(PDF), archived from the original (PDF) on 2013-01-23, retrieved 2012-09-07

- Jech, Thomas (2002).
*Set Theory*. Springer-Verlag. ISBN 3-540-44085-2.

- Media related to Subsets at Wikimedia Commons
- Weisstein, Eric W. "Subset".
*MathWorld*.

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