Subset

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Euler diagram showing
A is a subset of B (denoted
A
[?]
B
{\displaystyle A\subseteq B}
) and, conversely, B is a superset of A (denoted
B
[?]
A
{\displaystyle B\supseteq A}
). Venn A subset B.svg
Euler diagram showing
A is a subset of B (denoted ) and, conversely, B is a superset of A (denoted ).

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. A k-subset is a subset with k elements.

Contents

When quantified, is represented as [1]

One can prove the statement by applying a proof technique known as the element argument [2] :

Let sets A and B be given. To prove that

  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 for an arbitrarily chosen element c. Universal generalisation then implies which is equivalent to as stated above.

Definition

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 a proper (or strict) subset of B, denoted by , or equivalently,
  • B is a proper (or strict) superset of A, denoted by

The empty set, written or has no elements, and therefore is vacuously a subset of any set X.

Basic properties

A
[?]
B
{\displaystyle A\subseteq B}
and
B
[?]
C
{\displaystyle B\subseteq C}
implies
A
[?]
C
.
{\displaystyle A\subseteq C.} Subset with expansion.svg
and implies

Proper subset

⊂ and ⊃ symbols

Some authors use the symbols and to indicate subset and superset respectively; that is, with the same meaning as and instead of the symbols and [4] For example, for these authors, it is true of every set A that (a reflexive relation).

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 as and instead of the symbols and [5] This usage makes and analogous to the inequality symbols and For example, if then x may or may not equal y, but if then x definitely does not equal y, and is less than y (an irreflexive relation). Similarly, using the convention that is proper subset, if then A may or may not equal B, but if 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:

Power set

The set of all subsets of is called its power set, and is denoted by . [6]

The inclusion relation is a partial order on the set defined by . We may also partially order by reverse set inclusion by defining

For the power set of a set S, the inclusion partial order is—up to an order isomorphism—the Cartesian product of (the cardinality of S) copies of the partial order on for which This can be illustrated by enumerating , and associating with each subset (i.e., each element of ) the k-tuple from of which the ith coordinate is 1 if and only if is a member of T.

The set of all -subsets of is denoted by , in analogue with the notation for binomial coefficients, which count the number of -subsets of an -element set. In set theory, the notation is also common, especially when is a transfinite cardinal number.

Other properties of inclusion

See also

Related Research Articles

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References

  1. Rosen, Kenneth H. (2012). Discrete Mathematics and Its Applications (7th ed.). New York: McGraw-Hill. p.  119. ISBN   978-0-07-338309-5.
  2. Epp, Susanna S. (2011). Discrete Mathematics with Applications (Fourth ed.). p. 337. ISBN   978-0-495-39132-6.
  3. Stoll, Robert R. Set Theory and Logic. San Francisco, CA: Dover Publications. ISBN   978-0-486-63829-4.
  4. Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, p. 6, ISBN   978-0-07-054234-1, MR   0924157
  5. Subsets and Proper Subsets (PDF), archived from the original (PDF) on 2013-01-23, retrieved 2012-09-07
  6. Weisstein, Eric W. "Subset". mathworld.wolfram.com. Retrieved 2020-08-23.

Bibliography