§l. Set Theory
17
4 - Ordered pairs, Cartesian products, sets of subsets
If a and b are mathematical objects one has {a, b} = {b, a}. If, on the other
hand, you associate to every point of a plane furnished with coordinate axes
its two coordinates x and y and denote the corresponding point by the classical notation (x, y), it is clear that in general (x, y) :I (y, x). One is therefore
led to associate with two objects x and y written in a determinate order a
new object (x, y), an ordered pair, (or, in French, couple) the rule of equality
for two ordered pairs being that
(4.1)
(x,y) = (u,v) if and only if x = u and y = v.
Sometimes one says that x and yare the projections or coordinates of the
ordered pair (x, y). Similarly one defines triplets
(x,y,z) = «x,y),z),
quadruplets
(x, y, z, t) = «x, y, z), t),
etc. The rules of equality for such objects are obvious.
The axiom of pairs, used in n° 2 to define doubletons (non-ordered pairs)
{x, y}, allows one to introduce ordered pairs without stepping outside the
framework of Set Theory, by agreeing, for example, that
(4.2)
(x,y) = {{x},{x,y}},
an astute idea due to the Pole Kasimierz Kuratowski (1921). Suppes tells us it
was already to be found in another form in 1914 in the work of the American
Norbert Wiener, the "Father of Cybernetics", as journalists knowing nothing
else of him have called him since 1950. The relation (x, y) = (u, v), i.e.
(4.3)
{{x}, {x, y}} = {{u}, {u, v}},
in fact forces either {u} = {x}, or {u} = {x,y}. In the first case one has
u = x; in the second, x = y = u; so u = x in either case. If x = y, one
has {x,y} = {x}, hence {{x}, {x,y}} = {{x}} so that (3) can be written
{{x}} = {{x}, {x, v}}, which implies {x,v} = {x} i.e. v = x = y; if x:l y,
the second member of (3), which can be written {{x}, {x, v}} since u = x,
cannot contain the element {x,y} :I {x} unless {x,v} = {x,y}, and since
x :I y this forces v = y. In conclusion, one sees in every case that the
condition (1) is satisfied by the definition (2) of ordered pairs.
Exercise. Draw the boxes or graphs that represent the sets (2, 3) and (3, 2) .
Given two sets X and Y the set of ordered pairs (x, y) for which x E X
and y E Y is called the cartesian product of X and Y, and is denoted X x Y.
More generally one defines X x Y x Z = (X x Y) x Z, the set of triplets
(x, y, z) with x E X, Y E Z, z E Z, etc. If X is a set one defines
17
4 - Ordered pairs, Cartesian products, sets of subsets
If a and b are mathematical objects one has {a, b} = {b, a}. If, on the other
hand, you associate to every point of a plane furnished with coordinate axes
its two coordinates x and y and denote the corresponding point by the classical notation (x, y), it is clear that in general (x, y) :I (y, x). One is therefore
led to associate with two objects x and y written in a determinate order a
new object (x, y), an ordered pair, (or, in French, couple) the rule of equality
for two ordered pairs being that
(4.1)
(x,y) = (u,v) if and only if x = u and y = v.
Sometimes one says that x and yare the projections or coordinates of the
ordered pair (x, y). Similarly one defines triplets
(x,y,z) = «x,y),z),
quadruplets
(x, y, z, t) = «x, y, z), t),
etc. The rules of equality for such objects are obvious.
The axiom of pairs, used in n° 2 to define doubletons (non-ordered pairs)
{x, y}, allows one to introduce ordered pairs without stepping outside the
framework of Set Theory, by agreeing, for example, that
(4.2)
(x,y) = {{x},{x,y}},
an astute idea due to the Pole Kasimierz Kuratowski (1921). Suppes tells us it
was already to be found in another form in 1914 in the work of the American
Norbert Wiener, the "Father of Cybernetics", as journalists knowing nothing
else of him have called him since 1950. The relation (x, y) = (u, v), i.e.
(4.3)
{{x}, {x, y}} = {{u}, {u, v}},
in fact forces either {u} = {x}, or {u} = {x,y}. In the first case one has
u = x; in the second, x = y = u; so u = x in either case. If x = y, one
has {x,y} = {x}, hence {{x}, {x,y}} = {{x}} so that (3) can be written
{{x}} = {{x}, {x, v}}, which implies {x,v} = {x} i.e. v = x = y; if x:l y,
the second member of (3), which can be written {{x}, {x, v}} since u = x,
cannot contain the element {x,y} :I {x} unless {x,v} = {x,y}, and since
x :I y this forces v = y. In conclusion, one sees in every case that the
condition (1) is satisfied by the definition (2) of ordered pairs.
Exercise. Draw the boxes or graphs that represent the sets (2, 3) and (3, 2) .
Given two sets X and Y the set of ordered pairs (x, y) for which x E X
and y E Y is called the cartesian product of X and Y, and is denoted X x Y.
More generally one defines X x Y x Z = (X x Y) x Z, the set of triplets
(x, y, z) with x E X, Y E Z, z E Z, etc. If X is a set one defines