Page 31 - Advanced Linear Algebra

Preliminaries 15

Definition Let and denote cardinal numbers. Let and be disjoint sets
:

;
for which ((:~ and ( ( ~ ; .
1 The sum b is the cardinal number of r : . ;
)
)
2 The product is the cardinal number of :d ; .
3 The power is the cardinal number of : ; .
)
We will not go into the details of why these definitions make sense. For
(
)
instance, they seem to depend on the sets and , but in fact they do not. It
:
;
can be shown, using these definitions, that cardinal addition and multiplication
are associative and commutative and that multiplication distributes over
addition.
Theorem 0.13 Let , and be cardinal numbers. Then the following

properties hold:
1)(Associativity )

b ² b³ ~ ² b³ b and ² ³ ~ ² ³

2)(Commutativity )
b ~ b and ~
3)(Distributivity )

²b³ ~ b
4 )( Properties of Exponents)
b
~
a )
b ² ) ~
³

c ) ² ³ ~
On the other hand, the arithmetic of cardinal numbers can seem a bit strange, as
the next theorem shows.
Theorem 0.14 Let and be cardinal numbers, at least one of which is

infinite. Then
b ~ ~ ¸ max Á ¹
It is not hard to see that there is a one-to-one correspondence between the power
set F²:³ of a set and the set of all functions from to ¸ Á ¹ . This leads to
:
:
the following theorem.
Theorem 0.15 For any cardinal
)
(
F
1 If ((:~ , then ( ²:³ ~

2)