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CARTESIAN PRODUCT OF SETS
INTRODUCTION
When a beautiful Blondie
wants to choose what to wear for a date, she does not usually think about
Cartesian product of sets, but she might. Lets discuss all the
possibilities, if she has two skirts and two blouses. If we denote one
skirt with letter A, the other with letter B, one blouse with letter C and
another blouse with letter D, then she can look like this:
(A,C), (A,D), (B,C), (B,D).
If we say that A=1, B=2, C=3 and D= 4 we can draw this combination ina Cartesian coordinate
system.
Lets look at the Cartesian
coordinate system. In the plane U we have one vertical line and one
horizontal line that cross each other at point O. Each point is
represented by its coordinates, (a,b), what means that vertical line
through particular point crosses the horizontal axis at a and the
horizontal line through that particular point meets the vertical axis at
b. U×U is Cartesian plane.
Ordered pairs
An ordered pair consists of
two elements, for example a and b, in which one of them, let it be a, is
designated as the first element and the other as the second element. An
ordered pair is denoted by (a, b). Two ordered pairs (a, b) and (c,d) are
equal if and only if a = c and
b = d.
To the each point T U in the Cartesian coordinate system an exact ordered
pair (a,b) of real numbers is assigned.
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PRODUCT SET
Consider that we have two sets A and B. Let us take a look
at their product. The product
set of A and B consists of all ordered pairs (a, b) where a A and b B. It is denoted by A × B which
reads “A cross B”. A × B = {(a, b) | a A, b B}
The product set A × B is also called the Cartesian product of A and B.
Note that A × = ; × B =; × = ; A × B B × A, AB.
Example 1. Let
A={1, 2, 3 }
and B= {a, b, c, d}, then
A x B = {(1, a), (1,
b), (1, c), (1, d), (2, a), (2, b), (2, c), (2, d), (3, a), (3, b), (3,
c), (3, d)}
The function has its graph.
Consider that we have function f that maps A into B. Let
D(f)  A be the place where
the function f is defined. The graph f* of the function f consists of all
ordered pairs in which aD(f) appears as the first element and its image
appears as its second element.
f* = {(a,b) | a D(f), b = f(a)}
f : A  B  A×B.
Let us take a look at the properties of the graph of a
function. For that purpose let f be a function on two non empty sets A and
B. Let D(f) A be the place where
the function f is defined. Then for each element a D(f) there is assigned
an element in B. There is only one element in B which is assigned to each
a D(f).
The graph f* of f has the following two properties:
Property 1: For each a D(f), there is an
ordered pair (a, b) f*.
Property 2: Each a D(f) appears as the first element in only one ordered
pair in f*, that is,
(a,b) f*, (a,c)f*  b = c
If we talk about functions as sets of ordered pairs, then we
have to consider that f* A × B and let f* have the two properties:
Property 1: For each a D(f), D(f) is place
of f definition, there is an
ordered pair (a, b) f*.
Property 2: No two different ordered pairs in f*
have the same first element.
Definition: A function f of A into B is a subset of A × B in
which each a D(f), where D(f) is
place of f definition,
appears as the first element in one and only one ordered pair belonging to
f.
Generally talking about product sets we have to broaden our
considerations on more than two sets. If we have the Cartesian product of
three non empty sets A, B and C, A × B × C, then we take a deal with all
ordered triplets (a, b, c) where a A, b B and c C.
Analogously, the Cartesian product of n sets A1,
A2, …, An, denoted by A1 × A2
× … × An consists of all ordered n-tuples (a1,
a2, …, an) where a1 A1, a2
A2, …,
an An.
Axiom of Choice.
Let F be a collection of
nonempty sets. Then we can choose a member from each set in
that collection. Meanwhile, a function f is defined on F with the property that, for each
set S in the collection, f(S) is a member of S. The
Cartesian product of a non-empty family of non-empty sets is non-empty.
The function f is
then called a choice
function.
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