Thursday, March 7, 2013

Cardinal Numbers

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(A blog a week, keep my lazy away.... sort of)

Ordinal numbers are about order. Cardinal numbers are about quantities. In the previous post, we know what is an ordinal number. What is a cardinal number?
A cardinal number is a set of sets such that whenever we pick two sets in it, we can pair up their elements.  If two sets lie inside the same cardinal number, we say that they are of the same cardinality.
For example,  {a,b,c} and {x,y,z} are of the same cardinality, as we can pair up their elements as (a,x), (b,y) and (c,z).

A natural number is a cardinal number. Isn't it?
Well, we use a natural number as the representative of its cardinal number.
For example, {a,b,c} and 3={0,1,2} are of the same cardinality, and so we say that {a,b,c} has 3 elements.

So, if the elements of a set can be labelled as 0,1,2,...,n-1, then the set has n elements. Right?
Exactly.  It is just our usual sense of numbers. However, we are used to label the elements with 1,2,3,...,n instead.

The first infinite cardinal number is still ω={0,1,2,3,...} ?
Yeah.  Unlike ordinal numbers, 1+ω=ω=ω+1.
The sum of cardinal numbers are defined by, #A+#B=#(A×{0} ∪ B×{1}).
For example, 3+4=#{a,b,c}+#{x,y,z,w}=#{(a,0),(b,0),(c,0),(x,1),(y,1),(z,1),(w,1)}=7.

So, m+n=n+m?
Right. It is easy to see that A×{0} ∪ B×{1} and A×{1} ∪ B×{0} should have the same cardinality.

I guess the product of two cardinal numbers is (#A)(#B)=#(A×B)?
Good. For example 
3×4
=(#{a,b,c})(#{x,y,z,w})
=#{(a,x),(a,y),(a,z),(a,w),(b,x),(b,y),(b,z),(b,w),(c,x),(c,y),(c,z),(c,w)}
=12.
Again, we have m×n=n×m.