cardinal numbers 3

We will see a representative relationship lying between the cardinal number $\aleph_0$ of a countable set and $\aleph$ of an uncountable set.

Suppose a set $B$ be $\left\{0,1 \right\}$. For example, the cardinal number of the direct product set $B\times B\times B$ is card$(B\times B\times B)=2^3=8$, i.e.,
\[ 000, 001, 010, 011, 100, 101, 110, 111 \]
Hence, the cardinal number of the finite direct product set $B_1\times \cdots \times B_n$ is $2^n$.

We put a set $C$ as the infinite direct product set of $B$.
\[ C=\left\{ B\times B\times B\times \cdots   \right\} \]
card$C$ is expressed by $2^{\aleph_0}$ if $n\rightarrow \infty$. The set $C$ has all sequences of a infinite combinations of $0$ and $1$. In other words, $C$ contains all results of having thrown a coin with no end. We can show that $C$ is uncountable.

If $C$ is countable, all elements of $C$ can be listed in any order. However, after all elements were listed, by Cantor's diagonal argument, we can find a new sequence which should be in $C$. It is a contradiction. Therefore, $C$ must be uncountable. (refer to "countable sets 2")

This conclusion will be accepted by the simple fact that the binary number system is equivalent to the decimal in a scale of infinite digits.

As $C$ is uncountable, the cardinal number of $C$ becomes $\aleph$. However, since card$C$ is $2^{\aleph_0}$,
\[  2^{\aleph_0}=\aleph\]
It also means that
\[  \aleph_0 < 2^{\aleph_0} \]
Will you be convinced that it is true?

cardinal numbers 2

The cardinal number of an infinite set is very interesting. We are able to find some examples.

(1)Suppose $A_1=\left\{1,3,5,\cdots \right\}$ and $A_2=\left\{2,4,6,\cdots \right\}$. Then, obviously, card$A_1$=card$A_2$=$\aleph_0$. But, since $A_1\cup A_2=\mathbb{N}$, card$(A_1\cup A_2)=\aleph_0$. In the generality, for the direct sum of two infinite countable sets,
\[ \aleph_0+\aleph_0=\aleph_0 \]
(2)Similarly, in the cardinality of the continuum like the cardinal number of real numbers,
\[ \aleph +\aleph =\aleph \]
It can be understood by following facts. Let two sets be interval $A_1=(0,1]$ and $A_2=(1,2)$. The cardinal number card$A_1$=card$A_2$=$\aleph$. But, card$(A_1\cup A_2)=$card$(0,2)=\aleph$.

(3)We have already seen the open interval $(0,1)$ is one to one corresponding to $\mathbb{R}.  $Let a set $S$ be constructed by all points in a rectangle of the length $a>0$ and width $b>0$. Hence, $S$ is written by
\[ S=\left\{ (x,y)|x\in [0,a], y\in [0,b], a,b\in \mathbb{R}, a,b>0  \right\} \]
Do not confuse the direct product $(x,y)$ with an open interval.

We are able to arrange a one to one function $(x,y)\in S\rightarrow z\in\mathbb{R}$. Supposing $a=b=1$ and $z\in [0,1]$, we shall simplify the problem to understand. Let $c_i$ be a single digit figure from $0$ to $9$. If we define $x,y,z$ as follow,

$x=0.c_1c_3c_5\cdots $
$y=0.c_2c_4c_6\cdots $
$z=0.c_1c_2c_3c_4c_5c_6\cdots $

this function satisfies the conditions. That is, the function $(x,y)\rightarrow z$ is a one to one correspondence. Therefore, the cardinal number of $S$ is $\aleph$. It means in a direct product set that the following equation is true.
\[ \aleph^2=\aleph \]
Furthermore, in n-dimensions,
\[ \aleph^n=\aleph \]
You should note that a deciding factor of the cardinal number of a set is the existence of a one to one function. Will your intuition say yes?

coffee break 2

Almost in February/2013, I have enjoyed following entertainments.

[Translated Novels]
(1)☆☆Stark's War, Stark's Command, Stark's Crusade (John G. Hemry)
(2)☆☆Double play (Robert B. Parker)

[Japanese Novels]
(3)☆☆☆☆Sekai de ichiban shiawase na Okujyo, Bolero (On Yoshida)
(4)☆☆☆☆Uyoku to iu shokugyo (Hiroshi Take)
(5)☆☆Taitei no ken 4 (Baku Yumemakura) - to be continued
(6)☆☆☆Genso Yubin kyoku (Asako Horikawa)
(7)☆☆Neko machi (Sakutaro Hagiwara, Etsuko Kanai)

[Movies]
(8)☆☆Snatch (Jason Statham, Brad Pitt)
(9)☆☆Kokuriko zaka kara (animation, Ghibli)
(10)☆☆Metro ni notte (Shinichi Tutumi, Aya Okamoto)

In this month, Japanese novels are much fine. (3) is the sequel to the book "Yoru ni neko ga mi wo hisomeru tokoro, Think ". The latest is more excellent. The story is written well-carefully and the text is very stylish. We are surprised that the author On Yoshida is a teenager in addendum. However, she is actually a fictional author.

(4) is written by a former right-winger cadre. The story said to be a non-fiction shows much extremities by which I have been taken back. Those are very terrible and scare me.

cardinal numbers

The number of elements of a set is said to be the cardinal number. The cardinal number of a set $A$ is written by $\mbox{card}A$. If a set is finite, the cardinal number of the set becomes the number of elements. That is, if $A=\left\{-1,2,\sqrt{3},4,15 \right\}$, then $\mbox{card}A=5$.

Of course, $\mbox{card}\left\{0,1 \right\}=2$. We are able to define the cardinal number of the direct product of sets, too. If $A=\left\{0,1 \right\}\times \left\{0,1 \right\}\times \left\{0,1 \right\}$, or $A=\left\{(x_1, x_2, x_3)|x_i\in \left\{0, 1\right\}, i=1, 2, 3 \right\}$, then $\mbox{card}A=2^3=8$. If a set is finite, it is very simple.

In the case of an infinite set, you may not to be able to define the cardinal number, or may think that there is just only one cardinal number which is defined. However, we have understood an infinite set could be countable or uncountable. Hence, there are two types of infinite sets.

We define $\aleph_0$ as the cardinal number of a countably infinite set. For example,
\[ \mbox{card}\mathbb{Q}=\aleph_0  \]
Next, we define $\aleph $ as the cardinal number of real numbers.
\[ \mbox{card}\mathbb{R}=\aleph  \]
As real numbers are rational numbers plus irrational numbers, the number of elements of real numbers is more than those of rational numbers. Therefore,
\[ \aleph_0<\aleph \]
It was an astonishing fact discovered by G. Cantor that the number of elements of an infinite set also had a magnitude relationship.