In Mathématiques, a Ensemble has is a subset or a left a Ensemble B , or B is superset of has , if all element of the subset has is also element of the superset B . There can on the other hand be elements of B which are not elements of has (see the diagram on the right). The relation between has and B is called the inclusion .

Definitions

Inclusion, subsets and supersets

Are two A units and B. By definition, A is included in B if any element of A is an element of B. In notation symbolic system, inclusion is generally noted “⊂”. One has then by definition (“⇒” indicates the logical Implication):

HAS ⊂ B     mean     ∀ X ( X ∈ has ⇒ X ∈ B ).

Consequently the unit A is not included as a whole B if and only if there exists an element of A which does not belong to B:

HAS ⊄ B     if and only if     ∃ X ( X ∈ has and X ∉ B ).

For example the whole of the nonnull natural entireties N^* is included in the whole of the natural entireties NR , just as the whole of the even natural entireties 2 NR , but 2 NR is not included in N^* because 0 ∈ 2 NR , but 0 ∉ N^* :

N^* ⊂ NR , 2 NR ⊂ NR , 2 NR ⊄ N^* .

One can notice that, as there exist nonnull natural entireties which are not even, 1 for example, N^* is not included either in 2 NR : N^* ⊄ 2 NR . It is said whereas these two units are not comparable for inclusion .

Inclusion can be said in several ways, “A ⊂ B” can be also read:

• “A is contained in B”,
• “A is a left B”,
• or “A is a subset of B”.

and can be also written “B ⊃ A”, which is read:

• “B includes A”,
• “B contains A”,
• “B is a extension of A”,
• or “B is a superset of A”.

It is necessary to however take guard with the use of the term “contains” which is ambiguous, it can sometimes refer to the membership: has contains X can sometimes mean that has ∋ X (i.e. X ∈ has ).

Definition in comprehension

A property of the elements of a unit defines a subset of this one. Thus, by taking again one of the examples above, the property “being even” defines, on the whole of the natural entireties NR , unit 2 NR of the even entireties. It is said that the unit was defined by comprehension and one notes:

2 NR = { N ∈ NR | N is even} = { N ∈ NR | (∃ Q ∈ NR ) N =2 Q }

Any property (when one expresses it in a precise language one speaks about predicate of this language) defines by comprehension a subset of a given unit.

Strict inclusion and clean subsets

Let us notice that a unit is always subset of itself (see proposal 2 below). It can be necessary to exclude this case and to consider only subsets different from the unit itself. This is why one defines a strict inclusion , noted “⊊”. A unit has is strictly included in a unit B if and only if has is included in B without him to be equal:

$A\; \backslash subsetneq\; B$   mean     $A\; \backslash \; subset\; B$ and $A\; \backslash \; neq\; B$

Usual inclusion can then be described as broad inclusion , if there is risk of ambiguity.

Besides itself, a unit counts always at least another subset: the Empty set. These two subsets are sometimes known as “commonplace”. By opposition, the other subsets are called clean subsets (one says also left clean ).

Thus, by taking again the example of the preceding paragraph, the whole of the even natural entireties 2 NR , like the whole of the nonnull natural entireties N^* , is clean subsets of the whole of the natural entireties NR .

Together parts

The set of all the subsets of a unit E given is called together parts of E , and is usually noted “$\backslash \; mathcal\; P$ ( E )”, or (Gothic script) “$\backslash \; \_\; \backslash \; mathfrak\; P$ ( E )”, even simply “ P ( E )” (to read in all the cases “ P of E ”).
One has as follows:

X ∈ $\backslash \; mathcal\; P$ ( E )   if and only if   X ⊂ E .

For example if has = { has , B }, then $\backslash \; mathcal\; P$ ( has ) = {Ø, { has }, { B }, has }.

In this case there will be thus for example has ∈ has , and { has } ⊂ has .

The properties of all the parts, in particular those having milked with the cardinality, are detailed in the article Ensemble parts of a unit. For the case finished, which concerns the Combinatoire, also seeing the article Combinaison.

Characteristic function

A subset has of a unit E can be defined by its characteristic function   $\backslash chi\_A$ $\backslash \; \_\{\; :\; \backslash \; E\; \backslash \; rightarrow\; \backslash \; \{0,\; 1\; \backslash \}\}$, defined by χ_A ( X ) is worth 1 if X is element of has , and 0 if not:

$\backslash \; forall\; X\; \backslash \; in\; E\; \backslash \; chi\_A\; (X)\; =\; 1\; \backslash \; Leftrightarrow\; X\; \backslash \; in\; has$

and thus (χ_A being with values in {0,1})

$\backslash \; forall\; X\; \backslash \; in\; E\; X)\; =\; 0\; \backslash \; Leftrightarrow\; X\; \backslash \; not\; \backslash \; in\; has$

Reciprocally any function χ of E in {0,1} defines a subset of E which is { X ∈ E | χ ( X ) =1}. There is thus a bijective correspondence between the subsets of E and the functions of E in {0,1}, i.e. between $\backslash \; \_\; \backslash \; mathcal\; P$ ( E ) and {0,1} ^E .

Properties of inclusion

The Empty set is the unit which does not have elements, and it is noted Ø.

Proposal (empty set) . The Empty set is subset of any unit, i.e. for any unit has :

∅ ⊂ has

Demonstration: we must show that Ø is a subset of A, i.e. that all the elements of Ø are elements of A, but there do not exist elements of Ø. For which has a little the practice of mathematics, the inference “Ø does not have elements, therefore all the elements of Ø are elements of A” is obvious, but that can be disturbing for the beginner. He can be useful to reason differently (by the absurdity). If we had supposed that Ø was not a subset of A, we could have found an element of Ø not belonging to A. As there does not exist element of Ø, it is impossible and thus Ø is consequently a subset of A.

We have also the following proposal.

Proposal (reflexivity) . Any unit is included in itself, i.e. for any unit has :

HAS ⊂ HAS .

It is said that inclusion is a reflexive . To prove it, it is enough to take again the definition of inclusion.

Another property which it also rests only on the definition of inclusion is the transitivity.

Proposal (transitivity) . For three unspecified units A, B and C, if A is a subset of B and B is a subset of C, then A is a subset of C, i.e.:

( has ⊂ B and B ⊂ C ) ⇒ has ⊂ C .

Contrary to the preceding proposals, which show in a purely logical way, while returning to the definitions, the property of antisymetry rests on the even overall concept: it is in fact the simple translation of a fundamental property of the units, known as property of extensionnality, namely that two units are equal if and only if they have the same elements.

Proposal (antisymetry) . Two A units and B are equal if and only if A is a subset of B and B is a subset of A, i.e.:

HAS = B   if and only if   ( has ⊂ B and B ⊂ has )

Whatever the unit E , inclusion thus provides its together with the parts $\backslash \; mathcal\; P$ ( E ) of a Relation of order, which is a partial order as soon as E has at least two elements. Indeed if has and B is two elements distinct from E , singletons them { has } and { B } is parts of E which are not compared for inclusion. This order always has a Plus small element, Ø the empty set, and a Plus large element, the unit E .

This order is thus not total in general but has other remarkable properties.

Proposal (finished intersection) . For two units has and B unspecified, one can define the intersection has and B , which is the whole of the elements common to has and with B , noted has ∩ B . This unit is the only one with being included in has and in B , and to contain any unit contained at the same time in and in B has:

HAS ∩ B ⊂ HAS     and     HAS ∩ B ⊂ B ;

if C ⊂ has and C ⊂ B , then C ⊂ has ∩ B .

It is said that the unit has ∩ B is the lower Borne of has and B for inclusion.

There is a similar property (one says dual, in a precise direction) for the meeting.

Proposal (finished meeting) . For two units has and B unspecified, one can define the meeting has and B , which is the whole of the elements belonging to has or with B , noted has ∪ B . This unit is the only one to contain at the same time has and B , and with being contained in any unit containing at the same time has and B :

HAS ⊂ HAS ∪ B     and     B ⊂ HAS ∪ B ;

if has ⊂ C and B ⊂ C , then has ∪ B ⊂ C .

It is said that has ∪ B is the higher Borne of has and B for inclusion.

For any unit E inclusion thus provided $\backslash \; mathcal\; P$ ( E ) with a structure of order which one calls a lattice. One can define the inclusion starting from the intersection or of the meeting (it is a property common to the lattices):

HAS ⊂ B   if and only if   HAS ∩ B = HAS ;

HAS ⊂ B   if and only if   HAS ∪ B = B .

Because of the properties of Distributivité of the meeting with respect to the intersection, and of the intersection with respect to the meeting, this lattice is known as distributive.

Properties of the intersections and meetings binary, one could easily deduce a similar result for the intersections and finished meetings, but there is a stronger result:

Proposal (unspecified intersection and meeting) . For a unspecified family of units ( A_i ) _{I ∈ I} , one can define the intersection of the elements of the family, ∩ _{I ∈ I} A_i , and their meeting ∪ _{I ∈ I} A_i . The intersection of the A_i is largest of the units included in each A_i , the meeting of the A_i is smallest of the units including all the A_i .

The lattice of inclusion on $\backslash \; mathcal\; P$ ( E ) is known as complete. It is even about a Boolean algebra, since any subset of E has complementary in E .

Proposal (complementary) . That is to say E a unit. One will call Complémentaire of a subset has E , the subset of E consisted of the elements of E which are not in has , and one will note it has ^c. There a:

is ∩ has ^c = ∅   and   HAS ∪ HAS ^C = E

It is shown whereas:

HAS ⊂ B     if and only if     B ^C ⊂ HAS ^C

Axiomatic set theory

In axiomatic Set theory, in the Set theory of Zermelo or Zermelo-Fraenkel, inclusion is not a primitive concept. it is defined starting from the membership as indicated in the beginning of the article. As already mentioned, of the properties of inclusion, like reflexivity and transitivity, are consequences purely logical of this definition and the antisymetry of inclusion is exactly the Axiome of extensionnality. The existence of a smaller element (empty set) is shown by comprehension (see Axiome of the empty set). There is not a larger element for inclusion in the universe of the set theory: a unit which would contain all the units (within the meaning of inclusion) would be, by the Axiome of the pair (case of the singleton), the whole of all the units and one could, by using the Schéma of axioms of comprehension, to derive the Paradoxe from Russell. The existence of a lower Borne (intersection) is shown by comprehension. The existence of an upper limit (meeting), requires a specific axiom, the Axiome of the meeting. Each time the Axiome of extensionnality is useful to show unicity.

The existence of the whole of the parts of a unit also requires a specific axiom, the Axiome of the whole of the parts, and its unicity is once again ensured by the Axiome of extensionnality.

The membership and inclusion are in general quite distinct in ordinary mathematics. In set theory a very useful concept is that of Ensemble transitive: a unit of which all the elements are also subsets! In particular the ordinal are transitive units. The restriction of inclusion on ordinal defines a Bon order (and thus a total order), the strict order correspondent is the membership.

If one introduces, informellement or not, the concept of class (see the article corresponding), as this one corresponds to the concept of predicate, one can define in a similar way inclusion between classes. The class of all the units is maximum for inclusion. One can define the intersection and the meeting of two classes, and thus of a finished number of classes by Conjonction and Disjonction, the passage to complementary, by negation. The complementary one to a whole in a clean class, in particular in the class of all the units, cannot however be a unit (by meeting). It is not question on the other hand either overall, or even of class, the parts of a clean class.

See too

• Set theory

• overall Notion
• Opérations on the units
• Produces Cartesian
• Correspondances and Relations

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