1.NounAn 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."