Subset

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Euler diagram showing
A is a proper subset of B,  A[?]B,  and conversely B is a proper superset of A. Venn A subset B.svg
Euler diagram showing
A is a proper subset of B,  AB,  and conversely B is a proper superset of A.

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.

Contents

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.

Definitions

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, AB is represented as x(xAxB). [4]

We can prove the statement AB by applying a proof technique known as the element argument [5] :

Let sets A and B be given. To prove that AB,

  1. suppose that a is a particular but arbitrarily chosen element of A,
  2. show that a is an element of B.

The validity of this technique can be seen as a consequence of Universal generalization: the technique shows cAcB for an arbitrarily chosen element c. Universal generalisation then implies x(xAxB), which is equivalent to AB, as stated above.

Properties

Formally:
Formally:
Formally:

⊂ and ⊃ symbols

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 AA.

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 xy, 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 AB, then A may or may not equal B, but if AB, then A definitely does not equal B.

Examples of subsets

The regular polygons form a subset of the polygons PolygonsSet EN.svg
The regular polygons form a subset of the polygons

Another example in an Euler diagram:

Other properties of inclusion

A [?] B and B [?] C implies A [?] C Subset with expansion.svg
AB and BC implies AC

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 ab 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 = {s1, s2, ..., sk}, and associating with each subset TS (i.e., each element of 2S) the k-tuple from {0,1}k, of which the ith coordinate is 1 if and only if si is a member of T.

See also

Related Research Articles

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References

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

Bibliography