Subset - Wikipedia
Subset
"Superset" redirects here. For other uses, see Superset (disambiguation).
"⊃" redirects here. For the logic symbol, see horseshoe (symbol). For other uses, see horseshoe (disambiguation).
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.
Euler diagram showing
A is a proper subset of B,  AB,  and conversely B is a proper superset of A.
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:
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:
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
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 ⊇.[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
Another example in an Euler diagram:
Other properties of inclusion
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
References
  1. ^ a b c d "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
Jech, Thomas (2002). Set Theory. Springer-Verlag. ISBN 3-540-44085-2.
External links
Last edited on 21 March 2021, at 17:13
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