8
I - Sets and Functions
two relations E and = obey axioms which we shall state gradually as needed;
they permit us to construct ever more complex sets and relations.
The first, the axiom of extension, says that two sets A and B are equal
(Le., for the mathematician, identical or indistinguishable) if and only if they
possess the same elements; in other words, if the relations x E A and x E B
are logically equivalent. In the interpretation of a set X as a nesting of boxes
it is thus pointless to place in the primary box X two secondary boxes A and
B which, as nestings of boxes, have exactly the same structure: they are not
mathematically distinct objects even if physically they appear to be so. If for
example you place three empty boxes in a box X (or, more generally, three
copies of the same box), you obtain the same set as if you had placed only
one, since the axiom of extension shows that the two sets are mathematically
identical.
Given two sets X and Y one says that X is contained in Y (or that Y
contains X, or that X is a subset of Y) if every element of X is an element
of Y; the notation is Xc Y or Y :J X. It is clear that if Xc Y and Y c Z,
then X c Z. The relation X = Y means that both X c Y and Y eX.
Some authors write X ~ Y to maintain the visual analogy with x :::; y,
and reserve X c Y to mean that X is a proper subset of Y. This is a totally
useless notation.
If one considers, as has just been suggested, that mathematics consists
essentially of proving theorems about more or less complex nestings of boxes,
the simplest box one can imagine is the empty box. One thus needs a particular mathematical object, denoted 0, the empty set; its existence is the second
of the axioms of set theory12: there exists a set 0 such that the relation x E 0
is false for every x.
One meets this set in everyday life. If, when you are travelling by car
in the Far West, the police arrest you because you have shot a red light at
the intersection of two fiat, deserted, absolutely straight, orthogonal roads
you might acknowledge that your infraction constituted a mortal danger to
the, empty, set of motorists visible within a range of ten miles. (You would
pay the same fine, even so.) On 12 August 1997, the atmospheric pollution
in Paris having attained too high a level, the Parisian police, relayed by the
media, generously announced that Paris residents parking their cars within
their authorised perimeter were graciously absolved, the following day, from
paying their daily tribute of 15 F for this right 13 ; but all the motorists who
12 Some logicians prefer to postulate the existence of a set; that of 0 follows immediately from the axiom of separation stated below.
13 The idea is to encourage Parisian motorists to travel to work by public transport.
The fact is that normally using your car instead of parking in your street allows
you to save 15 F for parking and two metro tickets, i.e. enough money to buy
three litres of petrol. The duty of paying 15 F or a fine of 75 F if you do not use
your car to get to work before 9 a.m. thus amounts to subsidising the polluters
and penalising those who use public transport. This is confirmed by the fact
I - Sets and Functions
two relations E and = obey axioms which we shall state gradually as needed;
they permit us to construct ever more complex sets and relations.
The first, the axiom of extension, says that two sets A and B are equal
(Le., for the mathematician, identical or indistinguishable) if and only if they
possess the same elements; in other words, if the relations x E A and x E B
are logically equivalent. In the interpretation of a set X as a nesting of boxes
it is thus pointless to place in the primary box X two secondary boxes A and
B which, as nestings of boxes, have exactly the same structure: they are not
mathematically distinct objects even if physically they appear to be so. If for
example you place three empty boxes in a box X (or, more generally, three
copies of the same box), you obtain the same set as if you had placed only
one, since the axiom of extension shows that the two sets are mathematically
identical.
Given two sets X and Y one says that X is contained in Y (or that Y
contains X, or that X is a subset of Y) if every element of X is an element
of Y; the notation is Xc Y or Y :J X. It is clear that if Xc Y and Y c Z,
then X c Z. The relation X = Y means that both X c Y and Y eX.
Some authors write X ~ Y to maintain the visual analogy with x :::; y,
and reserve X c Y to mean that X is a proper subset of Y. This is a totally
useless notation.
If one considers, as has just been suggested, that mathematics consists
essentially of proving theorems about more or less complex nestings of boxes,
the simplest box one can imagine is the empty box. One thus needs a particular mathematical object, denoted 0, the empty set; its existence is the second
of the axioms of set theory12: there exists a set 0 such that the relation x E 0
is false for every x.
One meets this set in everyday life. If, when you are travelling by car
in the Far West, the police arrest you because you have shot a red light at
the intersection of two fiat, deserted, absolutely straight, orthogonal roads
you might acknowledge that your infraction constituted a mortal danger to
the, empty, set of motorists visible within a range of ten miles. (You would
pay the same fine, even so.) On 12 August 1997, the atmospheric pollution
in Paris having attained too high a level, the Parisian police, relayed by the
media, generously announced that Paris residents parking their cars within
their authorised perimeter were graciously absolved, the following day, from
paying their daily tribute of 15 F for this right 13 ; but all the motorists who
12 Some logicians prefer to postulate the existence of a set; that of 0 follows immediately from the axiom of separation stated below.
13 The idea is to encourage Parisian motorists to travel to work by public transport.
The fact is that normally using your car instead of parking in your street allows
you to save 15 F for parking and two metro tickets, i.e. enough money to buy
three litres of petrol. The duty of paying 15 F or a fine of 75 F if you do not use
your car to get to work before 9 a.m. thus amounts to subsidising the polluters
and penalising those who use public transport. This is confirmed by the fact