Laberinto

It is said that a Ensemble E is an even when it is made of two distinct elements has and B , and it is written then:

E= \ left \ {has, B \ right \}

When the unit E is made of only one element, has , it is said that E is a singleton , and it then is written:

E= \ left \ {has \ right \}

Remarks

One can use the notation { has , B } even if has = B . It is quite useful when variables are handled, one can write { X , there }, without supposing that X ≠ there , and that avoids multiplying the under-cases in a certain elementary number of reasoning of set theory. However if has = B then the unit { has , has } is not a pair but the singleton { has }.
the pair is a unit: she is not ordered. One can indifferently write { has , B } or { B , has } to indicate the same unit. This differentiates the pair from the couple.

{ has , B } = { B , has }, that has and B is or not distinct, while ( has , B ) ≠ ( B , has ) as soon as has and B is distinct.

Examples

$\ {1, 3 \}$ is a pair of Entier S.
$\ {sin, exp \}$ is a pair of functions to real variable.
$\ {\ {1 \}, \ {1, 2 \} \}$ is a pair of whole of entireties.

Properties

Membership of an element to a pair (or with a singleton)

One saw that the writing { has , B } can be used even if has = B . We will thus speak about even or singleton { has , B }.

An element X belongs to a pair if and only if it is equal to the one of the two elements of this pair. This statement is in fact as much valid for a singleton. One can thus write it formally, for has and B given:

∀ X '' X '' ∈ {'' has '', '' B ''} ⇔ ('' X '' = '' has '' or '' X '' = '' B '')

(it “or” in question indicates, as usual in mathematics, an inclusive disjunction: the statement remains true if X = has and X = B ).

This property characterizes the pairs (or let us singletons). When one axiomatizes the set theory, usually one uses a specific axiom, called Axiome of the pair, for the existence of a unit having this property, two elements not necessarily distinct being given.

In the case of let us singletons them ( has = B ), the index property can well-sure be simplified:

∀ X '' X '' ∈ {'' has ''} ⇔ '' X '' = '' has ''.

Equality of two pairs (or let us singletons)

Two pairs or let us singletons are equal if and only if their elements are equal two to two, in the two ways in which one can associate them. More precisely, for two pairs or let us singletons { has , B } and { C , D }:

'' B ''} = {'' C '', '' D ''} ⇔ = '' C '' and '' B '' = '' D '') or ('' has '' = '' D '' and '' B '' = '' C '')

Of course, the statement is simplified if it is known that one of the two units is a singleton:

= {'' C '', '' D ''} ⇔ = '' C '' and '' has '' = '' D ''

and is all the more simplified for the equality of singletons :

{ has } = { C } ⇔ has = C

Disjoined pairs

Two pairs or let us singletons are disjoined if and only if each element of the first pair or singleton is distinct from each of the two elements of the other, which, for pairs, means that the four elements of the two pairs are two to two distinct. For two pairs or let us singletons { has , B } and { C , D } one thus has:

{ has , B } ∩ { C , D } = ∅ if and only if ( has ≠ C and has ≠ D and B ≠ C and B ≠ D )

Cardinality of a pair

The cardinal of a unit is this one usually calls for the finished units his number of elements. A pair is thus obviously a finished whole of cardinal 2.

Other properties

a simple reasoning of Dénombrement watch that the number of pairs (“the true” pairs, without let us singletons) of a finished whole with N elements is equal to ${N (n-1) \ over 2}$ (see the article Combinaison).

See too

Axiom of the pair
Together
Singleton (mathematics)
Couple
Tuple

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