Noun
ordered pair (plural ordered pairs)
(set theory) An object containing exactly two elements in a fixed order, so that, when the elements are different, exchanging them gives a different object. Notation: (a, b) or
⟨
a
,
b
⟩
{\displaystyle \langle a,b\rangle }
.
If an ordered pair were defined (in terms of sets) as
(
x
,
y
)
:=
{
{
a
}
,
{
a
,
{
b
}
}
}
{\displaystyle (x,y):=\{\{a\},\{a,\{b\}\}\}}
then the "first element" of an ordered pair S could be defined as CAR(S) where CAR(S) = x if and only if
(
∀
y
∈
S
.
x
∈
y
)
{\displaystyle (\forall y\in S.\,x\in y)}
. Likewise, the "second element" of S could be defined as CDR(S) where CDR(S) = x if and only if
(
∃
y
∈
S
.
(
∃
z
∈
y
.
x
∈
z
)
)
{\displaystyle (\exists y\in S.\,(\exists z\in y.\,x\in z))}
. If the two elements happened to be equal, then the ordered pair would still have cardinality two as would be naturally expected.
Alternatively, the objects are called the first and second coordinates, or the left and right projections of the ordered pair. Source: Internet
Generally, an interval in mathematics corresponds to an ordered pair (x,y) taken from the direct product R × R of real numbers with itself. Source: Internet
An attribute is an ordered pair of attribute name and type name. Source: Internet
For an extensive discussion of the ordered pair in the context of Quinian set theories, see Holmes (1998). Source: Internet
For example, the ordered pair (0, 1), in the usual construction of the complex numbers with two-dimensional vectors. Source: Internet
For families of more than two sets, one may similarly replace each element by an ordered pair of the element and the index of the set that contains it. citation. Source: Internet