12
I - Sets and Functions
And X - X = 0 for any X.
If X and Yare two sets their intersection X n Y is the set of objects
belonging simultaneously to X and to Y; this definition is again legitimated
by the axiom of separation applied to X and the relation x E Y. One says
that X and Yare disjoint when X n Y = 0.
The union Au B of two sets is, intuitively, the set of objects belonging
to A or 17 to B. More generally, consider a set X and think of its elements
as themselves being sets; the axiom of union ·affirms the existence of a set
Y whose elements yare characterised by the following property: there exists
an A E X such that YEA; logicians call this the union of X, an expression
generally eschewed by mathematicians. It is the set of x such that x EE X.
In the imagery of boxes this signifies that one may suppress the various
secondary boxes belonging to X and replace them by the tertiary boxes they
contain, eliminating double mentions as always.
One ought to transform all this into an exciting game, one might even
make a fortune by patenting it. Mr Gates' employees would immediately devise a multicoloured speaking version ("now find the union of the union of
the union") for multimedia computers. There would be several degrees of difficulty, characterised by the maximum number of nestings allowed: the BIB
(Boxes in Boxes) for babies, the BIBIB (Boxes in Boxes in Boxes) etc., up to
BIBIBI ... (Boxes in Boxes in Boxes in ... ) at level ~o. One could, thanks
to the Internet, organise Olympiads on a planetary scale, as in mathematics.
The parents of future students of the Polytechnique, of Harvard, or of the
TodaY University in Tokyo, could present BIBIBIB to their offspring at the
age of six for the gifted, and three for the extra-gifted, the infinite ascensions
in the BIBIBI . .. being reserved for the Mozarts of Logic.
The axiom of union would allow one to legitimate logically the definition
of A U B if one knew that there was a set C of which A and B are elements.
The existence of C is as "evident" as is, in Euclidean geometry, the existence
of a unique straight line joining two given points. As evident naively, as
undemonstrable logically. We therefore need another axiom, the axiom of
pairs: if A and B are two sets there exists a set C of which A and B are the
only elements. C is unique, by the axiom of extension. One denotes it {A, B},
or {A} if A = B; you will have no trouble verifying that {A, B} = {B, A}.
Given three sets A, B, C one puts {A,B, C} ={A,B} U {C}, etc.
The operations of union and intersection have the following nearly obvious
properties:
XuY = YUX,
17 In mathematics the conjunction "or" is not disjunctive: if P and Q are statements, "P or Q" does not exclude "P and Q".
I - Sets and Functions
And X - X = 0 for any X.
If X and Yare two sets their intersection X n Y is the set of objects
belonging simultaneously to X and to Y; this definition is again legitimated
by the axiom of separation applied to X and the relation x E Y. One says
that X and Yare disjoint when X n Y = 0.
The union Au B of two sets is, intuitively, the set of objects belonging
to A or 17 to B. More generally, consider a set X and think of its elements
as themselves being sets; the axiom of union ·affirms the existence of a set
Y whose elements yare characterised by the following property: there exists
an A E X such that YEA; logicians call this the union of X, an expression
generally eschewed by mathematicians. It is the set of x such that x EE X.
In the imagery of boxes this signifies that one may suppress the various
secondary boxes belonging to X and replace them by the tertiary boxes they
contain, eliminating double mentions as always.
One ought to transform all this into an exciting game, one might even
make a fortune by patenting it. Mr Gates' employees would immediately devise a multicoloured speaking version ("now find the union of the union of
the union") for multimedia computers. There would be several degrees of difficulty, characterised by the maximum number of nestings allowed: the BIB
(Boxes in Boxes) for babies, the BIBIB (Boxes in Boxes in Boxes) etc., up to
BIBIBI ... (Boxes in Boxes in Boxes in ... ) at level ~o. One could, thanks
to the Internet, organise Olympiads on a planetary scale, as in mathematics.
The parents of future students of the Polytechnique, of Harvard, or of the
TodaY University in Tokyo, could present BIBIBIB to their offspring at the
age of six for the gifted, and three for the extra-gifted, the infinite ascensions
in the BIBIBI . .. being reserved for the Mozarts of Logic.
The axiom of union would allow one to legitimate logically the definition
of A U B if one knew that there was a set C of which A and B are elements.
The existence of C is as "evident" as is, in Euclidean geometry, the existence
of a unique straight line joining two given points. As evident naively, as
undemonstrable logically. We therefore need another axiom, the axiom of
pairs: if A and B are two sets there exists a set C of which A and B are the
only elements. C is unique, by the axiom of extension. One denotes it {A, B},
or {A} if A = B; you will have no trouble verifying that {A, B} = {B, A}.
Given three sets A, B, C one puts {A,B, C} ={A,B} U {C}, etc.
The operations of union and intersection have the following nearly obvious
properties:
XuY = YUX,
17 In mathematics the conjunction "or" is not disjunctive: if P and Q are statements, "P or Q" does not exclude "P and Q".