# Subset

Venn diagram : A is a (real) subset of B .

The mathematical terms subset and superset describe a relationship between two sets . Another word for subset is subset .

For the mathematical representation of the embedding of a subset to its basic amount mathematical function of the subset relationship that is inclusion mapping used. is a subset of and is a superset of if every element of is also contained in. If also contains further elements that are not contained in, then is a true subset of and is a true superset of . The set of all subsets of a given set is called the power set of . ${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle A}$

Georg Cantor - the "inventor" of set theory - coined the term subset from 1884; the symbol of the subset relation was introduced by Ernst Schröder in 1890 in his " Algebra of Logic ".

## definition

If and are sets and every element of is also an element of , one calls a subset or subset of : ${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B}$

${\ displaystyle A \ subseteq B: \ Longleftrightarrow \ forall x \ in A \ colon x \ in B}$

Conversely, the superset of is called if and only if is a subset of : ${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle B}$

${\ displaystyle B \ supseteq A: \ Longleftrightarrow A \ subseteq B}$

There is also the concept of the real subset. is a proper subset of if and only if a subset of and is not identical to . ${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B}$

${\ displaystyle A \ subsetneq B: \ Longleftrightarrow A \ subseteq B \ land A \ neq B}$

Again you write when . ${\ displaystyle B \ supsetneq A}$${\ displaystyle A \ subsetneq B}$

## Further notations

⊂⊊⊆⊇⊋⊃

Some authors also use the characters and for subset and superset instead of and . Most of the time, the author does not define the term “real subset”. ${\ displaystyle \ subset}$${\ displaystyle \ supset}$${\ displaystyle \ subseteq}$${\ displaystyle \ supseteq}$

Other authors prefer the characters and instead of and for proper subset and superset . This usage is fittingly reminiscent of the signs of inequality and . Since this notation is mostly used when the difference between real and non-real subset is important, the characters and are rarely used. ${\ displaystyle \ subset}$${\ displaystyle \ supset}$${\ displaystyle \ subsetneq}$${\ displaystyle \ supsetneq}$ ${\ displaystyle \ leq}$${\ displaystyle <}$${\ displaystyle \ subsetneq}$${\ displaystyle \ supsetneq}$

Variants of the character are also , and . If is not a subset of , can also be used. Corresponding spellings are for , and for , as well as (no superset). ${\ displaystyle \ subsetneq}$${\ displaystyle \ varsubsetneq}$${\ displaystyle \ subsetneqq}$${\ displaystyle \ varsubsetneqq}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A \ nsubseteq B: \ Longleftrightarrow \ lnot \ left (A \ subseteq B \ right)}$${\ displaystyle \ varsupsetneq}$${\ displaystyle \ supsetneq}$${\ displaystyle \ supsetneqq}$${\ displaystyle \ varsupsetneqq}$${\ displaystyle \ supsetneq}$${\ displaystyle A \ nsupseteq B}$

The corresponding Unicode symbols are: ⊂, ⊃, ⊆, ⊇, ⊄, ⊅, ⊈, ⊉, ⊊, ⊋ (see: Unicode Block Mathematical Operators ).

## Ways of speaking

Instead of “ is a subset of .”, “The amount is contained in the amount ” or “The amount is comprised of .” Is also used . In the same way, instead of “ is a superset of .”, “The set contains the set .” Or “The set includes the set .” Is also used. If there cannot be misunderstandings, “ contains ” etc. is also said. Misunderstandings can arise especially with "The quantity contains the element ." ${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B}$ ${\ displaystyle A}$

## Examples

The set {drum, playing card} is a subset of the set {guitar, playing card, digital camera, drum}
The regular polygons are a subset of the set of all polygons.
• {1, 2} is a (real) subset of {1, 2, 3}.
• {1, 2, 3} is a (spurious) subset of {1, 2, 3}.
• {1, 2, 3, 4} is not a subset of {1, 2, 3}.
• {1, 2, 3} is not a subset of {2, 3, 4}.
• {} is a (real) subset of {1, 2}.
• {1, 2, 3} is a (proper) superset of {1, 2}.
• {1, 2} is an (improper) superset of {1, 2}.
• {1} is not a superset of {1, 2}.
• The set of prime numbers is a real subset of the set of natural numbers .
• The set of rational numbers is a proper subset of the set of real numbers .

Further examples as quantity diagrams:

## properties

• The empty set is a subset of every set:
${\ displaystyle \ varnothing \ subseteq A}$
• Every set is a subset of itself:
${\ displaystyle A \ subseteq A}$
• Characterization of inclusion with the help of the association :
${\ displaystyle A \ subseteq B \ Leftrightarrow A \ cup B = B}$
• Characterization of inclusion using the average :
${\ displaystyle A \ subseteq B \ Leftrightarrow A \ cap B = A}$
• Characterization of inclusion using the difference set :
${\ displaystyle A \ subseteq B \ Leftrightarrow A \ setminus B = \ varnothing}$
• Characterization of inclusion using the characteristic function :
${\ displaystyle A \ subseteq B \ Leftrightarrow \ chi _ {A} \ leq \ chi _ {B}}$
• Two sets are equal if and only if each is a subset of the other:
${\ displaystyle A = B \ Leftrightarrow A \ subseteq B \ land B \ subseteq A}$
This rule is often used in proving the equality of two sets by showing mutual inclusion (in two steps).
• When transitioning to complement , the direction of inclusion reverses:
${\ displaystyle A \ subseteq B \ Rightarrow A ^ {\ rm {c}} \ supseteq B ^ {\ rm {c}}}$
• When forming the intersection, you always get a subset:
${\ displaystyle A \ cap B \ subseteq A}$
• When forming the union one always gets a superset:
${\ displaystyle A \ cup B \ supseteq A}$

## Inclusion as an order relation

If A ⊆ B and B ⊆ C, then A ⊆ C too

Inclusion as a relation between sets fulfills the three properties of a partial order relation , namely it is reflexive , antisymmetric and transitive :

${\ displaystyle A \ subseteq A}$
${\ displaystyle A \ subseteq B \ subseteq A \ Rightarrow A = B}$
${\ displaystyle A \ subseteq B \ subseteq C \ Rightarrow A \ subseteq C}$

(This is an abbreviation for and .) ${\ displaystyle A \ subseteq B \ subseteq C}$${\ displaystyle A \ subseteq B}$${\ displaystyle B \ subseteq C}$

So if there is a set of sets (a set system ), then there is a partial order . This is especially true for the power set of a given set . ${\ displaystyle M \,}$${\ displaystyle (M, \ subseteq)}$ ${\ displaystyle {\ mathcal {P}} (X)}$${\ displaystyle X}$

## Chains of inclusion

If a system of quantities is such that two of the quantities occurring in one encompass the other or are encompassed by the other, then such a system of quantities is called an inclusion chain . An example of this is provided by the system of left-hand unrestricted open intervals of . ${\ displaystyle M \,}$${\ displaystyle M \,}$${\ displaystyle \ {{] {- \ infty, x} [} \ mid x \ in \ mathbb {R} \}}$${\ displaystyle \ mathbb {R}}$

A special case of an inclusion chain is when a (finite or infinite) sequence of quantities is given, which is arranged in ascending or descending order. You then write briefly: ${\ displaystyle \ subseteq}$ ${\ displaystyle \ supseteq}$

${\ displaystyle A_ {1} \ subseteq A_ {2} \ subseteq A_ {3} \ subseteq \ ...}$
${\ displaystyle A_ {1} \ supseteq A_ {2} \ supseteq A_ {3} \ supseteq \ ...}$

## Size and number of subsets

• Every subset of a finite set is finite and the following applies to the cardinalities :
${\ displaystyle A \ subseteq B \ Rightarrow \ left | A \ right | \ leq \ left | B \ right |}$
${\ displaystyle A \ subsetneq B \ Rightarrow \ left | A \ right | <\ left | B \ right |}$
• Every superset of an infinite set is infinite.
• Even with infinite quantities, the following applies to the thicknesses:
${\ displaystyle A \ subseteq B \ Rightarrow \ left | A \ right | \ leq \ left | B \ right |}$
• In the case of infinite sets, however, it is possible that a real subset has the same cardinality as its basic set. For example, the natural numbers are a true subset of the whole numbers , but the two sets are equally powerful (namely, countably infinite ).
• After Cantor's theorem is the power set of a quantity always more powerful than the amount itself: .${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle | A | <{\ bigl |} {\ mathcal {P}} (A) {\ bigr |}}$
• A finite set with elements has exactly subsets.${\ displaystyle n}$${\ displaystyle 2 ^ {n}}$
• The number of -element subsets of a -element (finite) set is given by the binomial coefficient .${\ displaystyle k}$${\ displaystyle n}$ ${\ displaystyle {\ tbinom {n} {k}}}$

## literature

• Oliver Deiser: Introduction to set theory . Springer, 2004, ISBN 978-3-540-20401-5
• John L. Kelley: General Topology . Springer-Verlag, Berlin / Heidelberg / New York 1975, ISBN 3-540-90125-6 (reprint of the edition by Van Nostrand from 1955).

Wikibooks: Math for non-freaks: subset and real subset  - learning and teaching materials

## Individual evidence

1. a b Oliver Deiser: Introduction to set theory . Springer, 2004, ISBN 978-3-540-20401-5 , p. 33 ( excerpt (Google) ).
2. Adolf Fraenkel: Introduction to set theory: An elementary introduction to the realm of the infinitely great . Springer, 2nd edition, 2013, ISBN 9783662259009 , p. 15
3. Set theory . In: Encyclopedia of Mathematics .
4. Otto Kerner, Joseph Maurer, Jutta Steffens, Thomas Thode, Rudolf Voller: Vieweg Mathematik Lexikon . Vieweg, 1988, ISBN 3-528-06308-4 , p. 190.