Pair of mathematical objects
In
mathematics
, an
ordered pair
(
a
,
b
) is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (
a
,
b
) is different from the ordered pair (
b
,
a
) unless
a
=
b
. (In contrast, the
unordered pair
{
a
,
b
} equals the unordered pair {
b
,
a
}.)
Ordered pairs are also called
2-tuples
, or
sequences
(sometimes, lists in a computer science context) of length 2. Ordered pairs of
scalars
are sometimes called 2-dimensional
vectors
. (Technically, this is an abuse of
terminology
since an ordered pair need not be an element of a
vector space
.)
The entries of an ordered pair can be other ordered pairs, enabling the
recursive
definition of ordered
n
-tuples
(ordered lists of
n
objects). For example, the ordered triple (
a
,
b
,
c
) can be defined as (
a
, (
b
,
c
)), i.e., as one pair nested in another.
In the ordered pair (
a
,
b
), the object
a
is called the
first entry
, and the object
b
the
second entry
of the pair. Alternatively, the objects are called the first and second
components
, the first and second
coordinates
, or the left and right
projections
of the ordered pair.
Cartesian products
and
binary relations
(and hence
functions
) are defined in terms of ordered pairs, cf. picture.
Generalities
[
edit
]
Let
and
be ordered pairs. Then the
characteristic
(or
defining
)
property
of the ordered pair is:
The
set
of all ordered pairs whose first entry is in some set
A
and whose second entry is in some set
B
is called the
Cartesian product
of
A
and
B
, and written
A
×
B
. A
binary relation
between sets
A
and
B
is a
subset
of
A
×
B
.
The
(
a
,
b
)
notation may be used for other purposes, most notably as denoting
open intervals
on the
real number line
. In such situations, the context will usually make it clear which meaning is intended.
[1]
[2]
For additional clarification, the ordered pair may be denoted by the variant notation
, but this notation also has other uses.
The left and right
projection
of a pair
p
is usually denoted by
π
1
(
p
) and
π
2
(
p
), or by
π
ℓ
(
p
) and
π
r
(
p
), respectively.
In contexts where arbitrary
n
-tuples are considered,
π
n
i
(
t
) is a common notation for the
i
-th component of an
n
-tuple
t
.
Informal and formal definitions
[
edit
]
In some introductory mathematics textbooks an informal (or intuitive) definition of ordered pair is given, such as
For any two objects
a
and
b
, the ordered pair
(
a
,
b
)
is a notation specifying the two objects
a
and
b
, in that order.
[3]
This is usually followed by a comparison to a set of two elements; pointing out that in a set
a
and
b
must be different, but in an ordered pair they may be equal and that while the order of listing the elements of a set doesn't matter, in an ordered pair changing the order of distinct entries changes the ordered pair.
This "definition" is unsatisfactory because it is only descriptive and is based on an intuitive understanding of
order
. However, as is sometimes pointed out, no harm will come from relying on this description and almost everyone thinks of ordered pairs in this manner.
[4]
A more satisfactory approach is to observe that the characteristic property of ordered pairs given above is all that is required to understand the role of ordered pairs in mathematics. Hence the ordered pair can be taken as a
primitive notion
, whose associated axiom is the characteristic property. This was the approach taken by the
N. Bourbaki
group in its
Theory of Sets
, published in 1954. However, this approach also has its drawbacks as both the existence of ordered pairs and their characteristic property must be axiomatically assumed.
[3]
Another way to rigorously deal with ordered pairs is to define them formally in the context of set theory. This can be done in several ways and has the advantage that existence and the characteristic property can be proven from the axioms that define the set theory. One of the most cited versions of this definition is due to Kuratowski (see below) and his definition was used in the second edition of Bourbaki's
Theory of Sets
, published in 1970. Even those mathematical textbooks that give an informal definition of ordered pairs will often mention the formal definition of Kuratowski in an exercise.
Defining the ordered pair using set theory
[
edit
]
If one agrees that
set theory
is an appealing
foundation of mathematics
, then all mathematical objects must be defined as
sets
of some sort. Hence if the ordered pair is not taken as primitive, it must be defined as a set.
[5]
Several set-theoretic definitions of the ordered pair are given below( see also
[6]
).
Wiener's definition
[
edit
]
Norbert Wiener
proposed the first set theoretical definition of the ordered pair in 1914:
[7]
He observed that this definition made it possible to define the
types
of
Principia Mathematica
as sets.
Principia Mathematica
had taken types, and hence
relations
of all arities, as
primitive
.
Wiener used {{
b
}} instead of {
b
} to make the definition compatible with
type theory
where all elements in a class must be of the same "type". With
b
nested within an additional set, its type is equal to
's.
Hausdorff's definition
[
edit
]
About the same time as Wiener (1914),
Felix Hausdorff
proposed his definition:
"where 1 and 2 are two distinct objects different from a and b."
[8]
Kuratowski's definition
[
edit
]
In 1921
Kazimierz Kuratowski
offered the now-accepted definition
[9]
[10]
of the ordered pair (
a
,
b
):
When the first and the second coordinates are identical, the definition obtains:
Given some ordered pair
p
, the property "
x
is the first coordinate of
p
" can be formulated as:
The property "
x
is the second coordinate of
p
" can be formulated as:
In the case that the left and right coordinates are identical, the right
conjunct
is trivially true, since
Y
1
≠
Y
2
is never the case.
If
then:
This is how we can extract the first coordinate of a pair (using the
iterated-operation notation
for
arbitrary intersection
and
arbitrary union
):
This is how the second coordinate can be extracted:
(if
, then the set {y} could be obtained more simply:
, but the previous formula also takes into account the case when x=y)
Variants
[
edit
]
The above Kuratowski definition of the ordered pair is "adequate" in that it satisfies the characteristic property that an ordered pair must satisfy, namely that
. In particular, it adequately expresses 'order', in that
is false unless
. There are other definitions, of similar or lesser complexity, that are equally adequate:
- [11]
The
reverse
definition is merely a trivial variant of the Kuratowski definition, and as such is of no independent interest. The definition
short
is so-called because it requires two rather than three pairs of
braces
. Proving that
short
satisfies the characteristic property requires the
Zermelo?Fraenkel set theory
axiom of regularity
.
[12]
Moreover, if one uses
von Neumann's set-theoretic construction of the natural numbers
, then 2 is defined as the set {0, 1} = {0, {0}}, which is indistinguishable from the pair (0, 0)
short
. Yet another disadvantage of the
short
pair is the fact that, even if
a
and
b
are of the same type, the elements of the
short
pair are not. (However, if
a
=
b
then the
short
version keeps having cardinality 2, which is something one might expect of any "pair", including any "ordered pair".)
Proving that definitions satisfy the characteristic property
[
edit
]
Prove: (
a
,
b
) = (
c
,
d
)
if and only if
a
=
c
and
b
=
d
.
Kuratowski
:
If
. If
a
=
c
and
b
=
d
, then {{
a
}, {
a
,
b
}} = {{
c
}, {
c
,
d
}}. Thus (
a, b
)
K
= (
c
,
d
)
K
.
Only if
. Two cases:
a
=
b
, and
a
≠
b
.
If
a
=
b
:
- (
a, b
)
K
= {{
a
}, {
a
,
b
}} = {{
a
}, {
a
,
a
}} = {{
a
}}.
- {{
c
}, {
c
,
d
}} = (
c
,
d
)
K
= (
a
,
b
)
K
= {{
a
}}.
- Thus {
c
} = {
c
,
d
} = {
a
}, which implies
a
=
c
and
a
=
d
. By hypothesis,
a
=
b
. Hence
b
=
d
.
If
a
≠
b
, then (
a
,
b
)
K
= (
c
,
d
)
K
implies {{
a
}, {
a
,
b
}} = {{
c
}, {
c
,
d
}}.
- Suppose {
c
,
d
} = {
a
}. Then
c
=
d
=
a
, and so {{
c
}, {
c
,
d
}} = {{
a
}, {
a
,
a
}} = {{
a
}, {
a
}} = {{
a
}}. But then {{
a
}, {
a, b
}} would also equal {{
a
}}, so that
b
=
a
which contradicts
a
≠
b
.
- Suppose {
c
} = {
a
,
b
}. Then
a
=
b
=
c
, which also contradicts
a
≠
b
.
- Therefore {
c
} = {
a
}, so that
c = a
and {
c
,
d
} = {
a
,
b
}.
- If
d
=
a
were true, then {
c
,
d
} = {
a
,
a
} = {
a
} ≠ {
a
,
b
}, a contradiction. Thus
d
=
b
is the case, so that
a
=
c
and
b
=
d
.
Reverse
:
(
a, b
)
reverse
= {{
b
}, {
a, b
}} = {{
b
}, {
b, a
}} = (
b, a
)
K
.
If
. If (
a, b
)
reverse
= (
c, d
)
reverse
,
(
b, a
)
K
= (
d, c
)
K
. Therefore,
b = d
and
a = c
.
Only if
. If
a = c
and
b = d
, then {{
b
}, {
a, b
}} = {{
d
}, {
c, d
}}.
Thus (
a, b
)
reverse
= (
c, d
)
reverse
.
Short:
[13]
If
: If
a = c
and
b = d
, then {
a
, {
a, b
}} = {
c
, {
c, d
}}. Thus (
a, b
)
short
= (
c, d
)
short
.
Only if
: Suppose {
a
, {
a, b
}} = {
c
, {
c, d
}}.
Then
a
is in the left hand side, and thus in the right hand side.
Because equal sets have equal elements, one of
a = c
or
a
= {
c, d
} must be the case.
- If
a
= {
c, d
}, then by similar reasoning as above, {
a, b
} is in the right hand side, so {
a, b
} =
c
or {
a, b
} = {
c, d
}.
- If {
a, b
} =
c
then
c
is in {
c, d
} =
a
and
a
is in
c
, and this combination contradicts the axiom of regularity, as {
a, c
} has no minimal element under the relation "element of."
- If {
a, b
} = {
c, d
}, then
a
is an element of
a
, from
a
= {
c, d
} = {
a, b
}, again contradicting regularity.
- Hence
a = c
must hold.
Again, we see that {
a, b
} =
c
or {
a, b
} = {
c, d
}.
- The option {
a, b
} =
c
and
a = c
implies that
c
is an element of
c
, contradicting regularity.
- So we have
a = c
and {
a, b
} = {
c, d
}, and so: {
b
} = {
a, b
} \ {
a
} = {
c, d
} \ {
c
} = {
d
}, so
b
=
d
.
Quine?Rosser definition
[
edit
]
Rosser
(1953)
[14]
employed a definition of the ordered pair due to
Quine
which requires a prior definition of the
natural numbers
. Let
be the set of natural numbers and define first
The function
increments its argument if it is a natural number and leaves it as is otherwise; the number 0 does not appear as functional value of
.
As
is the set of the elements of
not in
go on with
This is the
set image
of a set
under
,
sometimes denoted
by
as well. Applying function
to a set
x
simply increments every natural number in it. In particular,
does never contain the number 0, so that for any sets
x
and
y
,
Further, define
By this,
does always contain the number 0.
Finally, define the ordered pair (
A
,
B
) as the disjoint union
(which is
in alternate notation).
Extracting all the elements of the pair that do not contain 0 and undoing
yields
A
. Likewise,
B
can be recovered from the elements of the pair that do contain 0.
[15]
For example, the pair
is encoded as
provided
.
In
type theory
and in outgrowths thereof such as the axiomatic set theory
NF
, the Quine?Rosser pair has the same type as its projections and hence is termed a "type-level" ordered pair. Hence this definition has the advantage of enabling a
function
, defined as a set of ordered pairs, to have a type only 1 higher than the type of its arguments. This definition works only if the set of natural numbers is infinite. This is the case in
NF
, but not in
type theory
or in
NFU
.
J. Barkley Rosser
showed that the existence of such a type-level ordered pair (or even a "type-raising by 1" ordered pair) implies the
axiom of infinity
. For an extensive discussion of the ordered pair in the context of Quinian set theories, see Holmes (1998).
[16]
Cantor?Frege definition
[
edit
]
Early in the development of the set theory, before paradoxes were discovered, Cantor followed Frege by defining the ordered pair of two sets as the class of all relations that hold between these sets, assuming that the notion of relation is primitive:
[17]
This definition is inadmissible in most modern formalized set theories and is methodologically similar to defining the
cardinal
of a set as the class of all sets equipotent with the given set.
[18]
Morse definition
[
edit
]
Morse?Kelley set theory
makes free use of
proper classes
.
[19]
Morse
defined the ordered pair so that its projections could be proper classes as well as sets. (The Kuratowski definition does not allow this.) He first defined ordered pairs whose projections are sets in Kuratowski's manner. He then
redefined
the pair
where the component Cartesian products are Kuratowski pairs of sets and where
This renders possible pairs whose projections are proper classes. The Quine?Rosser definition above also admits
proper classes
as projections. Similarly the triple is defined as a 3-tuple as follows:
The use of the singleton set
which has an inserted empty set allows tuples to have the uniqueness property that if
a
is an
n
-tuple and b is an
m
-tuple and
a
=
b
then
n
=
m
. Ordered triples which are defined as ordered pairs do not have this property with respect to ordered pairs.
Axiomatic definition
[
edit
]
Ordered pairs can also be introduced in
Zermelo?Fraenkel set theory
(ZF) axiomatically by just adding to ZF a new function symbol
of arity 2 (it is usually omitted) and a defining axiom for
:
This definition is acceptable because this extension of ZF is a
conservative extension
.
[
citation needed
]
The definition helps to avoid so called accidental theorems like (a,a) = {{a}}, {a} ∈ (a,b), if Kuratowski's definition (a,b) = {{a}, {a,b}} was used.
Category theory
[
edit
]
A category-theoretic
product
A
×
B
in a
category of sets
represents the set of ordered pairs, with the first element coming from
A
and the second coming from
B
. In this context the characteristic property above is a consequence of the
universal property
of the product and the fact that elements of a set
X
can be identified with morphisms from 1 (a one element set) to
X
. While different objects may have the universal property, they are all
naturally isomorphic
.
See also
[
edit
]
References
[
edit
]
- ^
Lay, Steven R. (2005),
Analysis / With an Introduction to Proof
(4th ed.), Pearson / Prentice Hall, p. 50,
ISBN
978-0-13-148101-5
- ^
Devlin, Keith (2004),
Sets, Functions and Logic / An Introduction to Abstract Mathematics
(3rd ed.), Chapman & Hall / CRC, p. 79,
ISBN
978-1-58488-449-1
- ^
a
b
Wolf, Robert S. (1998),
Proof, Logic, and Conjecture / The Mathematician's Toolbox
, W. H. Freeman and Co., p. 164,
ISBN
978-0-7167-3050-7
- ^
Fletcher, Peter; Patty, C. Wayne (1988),
Foundations of Higher Mathematics
, PWS-Kent, p. 80,
ISBN
0-87150-164-3
- ^
Quine
has argued that the set-theoretical implementations of the concept of the ordered pair is a paradigm for the clarification of philosophical ideas (see "
Word and Object
", section 53).
The general notion of such definitions or implementations are discussed in Thomas Forster "Reasoning about theoretical entities".
- ^
Dipert, Randall.
"Set-Theoretical Representations of Ordered Pairs and Their Adequacy for the Logic of Relations"
.
- ^
Wiener's paper "A Simplification of the logic of relations" is reprinted, together with a valuable commentary on pages 224ff in van Heijenoort, Jean (1967),
From Frege to Godel: A Source Book in Mathematical Logic, 1979?1931
, Harvard University Press, Cambridge MA,
ISBN
0-674-32449-8
(pbk.). van Heijenoort states the simplification this way: "By giving a definition of the ordered pair of two elements in terms of class operations, the note reduced the theory of relations to that of classes".
- ^
cf introduction to Wiener's paper in van Heijenoort 1967:224
- ^
cf introduction to Wiener's paper in van Heijenoort 1967:224. van Heijenoort observes that the resulting set that represents the ordered pair "has a type higher by 2 than the elements (when they are of the same type)"; he offers references that show how, under certain circumstances, the type can be reduced to 1 or 0.
- ^
Kuratowski, Casimir
(1921).
"Sur la notion de l'ordre dans la Theorie des Ensembles"
.
Fundamenta Mathematicae
.
2
(1): 161?171.
doi
:
10.4064/fm-2-1-161-171
.
- ^
This differs from Hausdorff's definition in not requiring the two elements 0 and 1 to be distinct from
a
and
b
.
- ^
Tourlakis, George (2003)
Lectures in Logic and Set Theory. Vol. 2: Set Theory
. Cambridge Univ. Press. Proposition III.10.1.
- ^
For a formal
Metamath
proof of the adequacy of
short
, see
here (opthreg).
Also see Tourlakis (2003), Proposition III.10.1.
- ^
J. Barkley Rosser
, 1953.
Logic for Mathematicians
. McGraw?Hill.
- ^
Holmes, M. Randall
:
On Ordered Pairs
, on: Boise State, March 29, 2009. The author uses
for
and
for
.
- ^
Holmes, M. Randall (1998)
Elementary Set Theory with a Universal Set
Archived
2011-04-11 at the
Wayback Machine
. Academia-Bruylant. The publisher has graciously consented to permit diffusion of this monograph via the web.
- ^
Frege, Gottlob (1893). "144".
Grundgesetze der Arithmetik
(PDF)
. Jena: Verlag Hermann Pohle.
- ^
Kanamori, Akihiro (2007).
Set Theory From Cantor to Cohen
(PDF)
. Elsevier BV.
p. 22, footnote 59
- ^
Morse, Anthony P. (1965).
A Theory of Sets
. Academic Press.