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In mathematics, especially in set theory, a set A is a subset of a set B, if A is "contained" inside B. The relationship of one set being a subset of another is called inclusion. Every set is a subset of itself. Venn diagram for A is a subset of B. Image created by myself File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Venn diagram for A is a subset of B. Image created by myself File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...

More formally, If A and B are sets and every element of A is also an element of B, then: In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). ...

• A is a subset of (or is included in) B, denoted by AB,

or equivalently

• B is a superset of (or includes) A, denoted by BA.

If A is a subset of B, but A is not equal to B, then A is also a proper (or strict) subset of B. This is written as AB. In the same way, BA means that B is a proper superset of A. See also the disambiguation page title equality. ...

An easy way to remember the difference in symbols is to note that ⊆ and ⊂ are analogous to ≤ and <. For example, if A is a subset of B (written as AB), then the number of elements in A is less than or equal to the number of elements in B (written as |A| ≤ |B|). Likewise, for finite sets A and B, if AB then |A| < |B|. In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...

There are other, equivalent interpretations, all of them important.

• If A is a subset of B, then $x in A rightarrow x in B$.
• If A is a proper subset of B, then there exists at least one element x such that $x in B$ but $x notin A$.

N.B. Many authors do not follow the above conventions, but use ⊂ to mean simply subset (rather than proper subset). There is an unambiguous symbol, $subsetneq$ (or in Unicode), for proper subset. Some authors use both unambiguous symbols, ⊆ for subset and $subsetneq$ for proper subset, and dispense with ⊂ altogether. The corresponding remarks apply for supersets as well. Due to technical limitations, some web browsers may not display some special characters in this article. ...

For any set S, inclusion is a relation on the set of all subsets of S (the power set of S). In mathematics, the concept of binary relation, sometimes called dyadic relation, is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ... In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ...

## Examples

• The set {1, 2} is a proper subset of {1, 2, 3}.
• The set of natural numbers is a proper subset of the set of rational numbers.
• The set {x : x is a prime number greater than 2000} is a proper subset of {x : x is an odd number greater than 1000}
• Any set is a subset of itself, but not a proper subset.
• The empty set, written ø, is also a subset of any given set X. (This statement is vacuously true, see proof below) The empty set is always a proper subset, except of itself.

In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... Informally, a logical statement is vacuously true if it is true but doesnt say anything; examples are statements of the form everything with property A also has property B, where there is nothing with property A. It is tempting to dismiss this concept as vacuous or silly. ...

## Properties

PROPOSITION 1: The empty set is a subset of every set. In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...

Proof: Given any set A, we wish to prove that ø is a subset of A. This involves showing that all elements of ø are elements of A. But there are no elements of ø.

For the experienced mathematician, the inference " ø has no elements, so all elements of ø are elements of A" is immediate, but it may be more troublesome for the beginner. Since ø has no members at all, how can "they" be members of anything else? It may help to think of it the other way around. In order to prove that ø was not a subset of A, we would have to find an element of ø which was not also an element of A. Since there are no elements of ø, this is impossible and hence ø is indeed a subset of A. Informally, a logical statement is vacuously true if it is true but doesnt say anything; examples are statements of the form everything with property A also has property B, where there is nothing with property A. It is tempting to dismiss this concept as vacuous or silly. ...

The following proposition says that inclusion is a partial order. In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ...

PROPOSITION 2: If A, B and C are sets then the following hold:

reflexivity:
• A ⊆ A
antisymmetry:
• A ⊆ B and B ⊆ A if and only if A = B
transitivity:
• If A ⊆ B and B ⊆ C then A ⊆ C

The following proposition says that for any set S the power set of S ordered by inclusion is a bounded lattice, and hence together with the distributive and complement laws for unions and intersections (see The fundamental laws of set algebra), show that it is a Boolean algebra. In set theory, a binary relation can have, among other properties, reflexivity or irreflexivity. ... In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X, if a is related to b and b is related to a, then a = b. ... In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ... In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... The algebra of sets develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality (mathematics) and set inclusion. ... Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ...

PROPOSITION 3: If A, B and C are subsets of a set S then the following hold:

existence of a least element and a greatest element:
• ø ⊆ A ⊆ S (that ø ⊆ A is Proposition 1 above.)
existence of joins:
• A ⊆ AB
• If A ⊆ C and B ⊆ C then AB ⊆ C
existence of meets:
• AB ⊆ A
• If C ⊆ A and C ⊆ B then C ⊆ AB

The following proposition says that, the statement "A ⊆ B ", is equivalent to various other statements involving unions, intersections and complements. In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually. ... In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...

PROPOSITION 4: For any two sets A and B, the following are equivalent:

• A ⊆ B
• A ∩ B  =  A
• A ∪ B  =  B
• A − B  =   ø
• B′ ⊆ A

The above proposition shows that the relation of set inclusion can be characterized by either of the set operations of union or intersection, which means that the notion of set inclusion is axiomatically superfluous.

## Other properties of inclusion

The usual order on the ordinal numbers is given by inclusion. Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...

For the power set of a set S, the inclusion partial order is (up to an order-isomorphism) the Cartesian product of |S| (the cardinality of S) copies of the partial order on {0,1}, for which 0 < 1. In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... In the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets. ... In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X Ã— Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y: The Cartesian product is named after RenÃ© Descartes... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality â€“ one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... Results from FactBites:

 Superset workout (946 words) Supersets and compound sets work very well for this and you can do the same number of bodyparts in half the normal time! A superset is two or more exercises in a row, then a rest break. There are some exercises to work the front of the calves but it's not really practical to try to superset them and for most people they're not necessary.
 PlanetMath: superset (156 words) Every set is a superset of itself, and every set is a superset of the empty set. This is version 9 of superset, born on 2002-02-24, modified 2006-09-17. Object id is 2585, canonical name is Superset.
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