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Subset

Subset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is contained in B. That is, all elements of A are also elements of B (note that A and B may be equal). 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 A, or A is included in B.

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 meet and join are given by intersection and union.

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

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 also a proper (or strict) subset of B; this is written as
or equivalently
  • B is a proper superset of A; this is written as
For any set S, the inclusionrelation⊆ is apartial orderon the setof all subsets of S (thepower setof S) defined by. We may also partially orderby reverse set inclusion by defining

When quantified, A ⊆ B is represented as: ∀x{x∈A → x∈B}.[1]

Properties

  • 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:

⊂ 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 ⊇.[2] 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 ⊋.[3] 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.

Examples

  • 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 empty set { }, denoted by ∅, is also a subset of any given set X. It is also always a proper subset of any set except itself.

  • 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

Other properties of inclusion

Inclusion is the canonicalpartial orderin the sense that every partially ordered set (X,) isisomorphicto some collection of sets ordered by inclusion. Theordinal numbersare 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 thepower setof a set S, the inclusion partial order is (up to anorder isomorphism) theCartesian productof k = |S| (thecardinalityof 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 T ⊆ S (which is to say with each element of 2S) the k-tuple from {0,1}kof which the ith coordinate is 1 if and only if siis a member of T.

See also

  • Containment order

References

[1]
Citation Linkopenlibrary.orgRosen, Kenneth H. (2012). Discrete Mathematics and Its Applications (7th ed.). New York: McGraw-Hill. p. 119. ISBN 978-0-07-338309-5.
Sep 30, 2019, 3:52 PM
[2]
Citation Link//www.ams.org/mathscinet-getitem?mr=0924157Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, p. 6, ISBN 978-0-07-054234-1, MR 0924157
Sep 30, 2019, 3:52 PM
[3]
Citation Linkit.edgecombe.eduSubsets and Proper Subsets (PDF), archived from the original (PDF) on 2013-01-23, retrieved 2012-09-07
Sep 30, 2019, 3:52 PM
[4]
Citation Linkmathworld.wolfram.com"Subset"
Sep 30, 2019, 3:52 PM
[5]
Citation Linkwww.ams.org0924157
Sep 30, 2019, 3:52 PM
[6]
Citation Linkweb.archive.orgSubsets and Proper Subsets
Sep 30, 2019, 3:52 PM
[7]
Citation Linkit.edgecombe.eduthe original
Sep 30, 2019, 3:52 PM
[8]
Citation Linkmathworld.wolfram.com"Subset"
Sep 30, 2019, 3:52 PM
[9]
Citation Linken.wikipedia.orgThe original version of this page is from Wikipedia, you can edit the page right here on Everipedia.Text is available under the Creative Commons Attribution-ShareAlike License.Additional terms may apply.See everipedia.org/everipedia-termsfor further details.Images/media credited individually (click the icon for details).
Sep 30, 2019, 3:52 PM