Infinite Sets

May 20, 2009 13:56 by scibuff
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I distinctly remember one day in second grade when the day after our first class on multiplication my friend infamously claimed that

\frac{1}{0}=\infty

Although he was definitely incorrect in his thinking and reasoning, because division be zero has no meaning when it comes to real numbers (or integers), there are cases, such as  The Riemann Sphere or a one-element field where the multiplicative identity coincides with the additive identity, when the equation above is actually true (and well defined).

It is sometimes quite tempting (and even useful) to consider

\lim_{x\to 0}\frac{1}{x}=\frac{\lim_{x\to 0}1}{\lim_{x\to 0}x}=\frac{1}{0}=\infty

The main problem with allowing the division by zero is that it results in logical fallacies such as 1=0 (actually, one can equate any two numbers using it). Eventually, mathematicians found a way to define limits rigorously without infinitesimal quantities needed by Newton and Leibnitz when Cauchy and Weierstrass laid down the foundation of modern analysis.

The next question from Yahoo Answers is related to a different kind of infinity – the kind that deals with sets. The works of Georg Cantor between 1874 and 1884 are the origin of set theory in which he established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are “more infinite” than the natural numbers. The diagonalization argument he used demonstrates a powerful and general technique that has since been used in a wide range of proofs such as this one.

Problem: Determine whether the set of real numbers between 0 and 1 with decimal representations consisting of 1s, i.e. S=\left \{0.1, 0.11, 0.111,\cdots \right \} is countable or uncountable?

Definitions:

  • A function f:A\rightarrow B is bijective if and only if for every b\in B there is exactly one a\in A such that f(a)=b
  • A set S is said to be countable if there exists a bijective function f:\mathbb{N}\rightarrow S where \mathbb{N}=\left \{0, 1, 2, 3,\cdots \right \} is the set of natural numbers.

Solution:

Let us look at set S=\left \{0.1, 0.11, 0.111,\cdots \right \} and consider the following list:

0:\ 0.1

1:\ 0.11

2:\ 0.111

\vdots

It is easy to see that our set S can be easily put into a one-to-one correspondence with the set of natural numbers \mathbb{N}=\left \{0, 1, 2, 3,\cdots \right \}, e.g. the correspondence is given by the list (function defined) above.

To be precise, let f:\mathbb{N}\rightarrow S be given by

f(n)=\overset{k=n}{\underset{k=0}{\sum}}10^{-(k+1)}

To show that f is bijective, we need to show that  for every s_{n}\in S there is exactly one, i.e. a unique, n\in \mathbb{N} such that f(n)=s_{n}.

Consider an arbitrary s_{m}\in S. Since

s_{m}=0.\overset{m}{\overbrace{111\cdots 111}}

we have

s_{m}=0.1+0.01+\cdots+0.\overset{m\ 0s}{\overbrace{000\cdots 000}}1 =10^{-1}+10^{-2}+\cdots + 10^{-(m+1)}

and so by taking n=m we get

s_{m}=10^{-1}+10^{-2}+\cdots + 10^{-(m+1)}=\overset{k=m}{\underset{k=0}{\sum}}10^{-(k+1)}=f(m)

Therefore, for every s_{n}\in S we have proven the existance of (at least one) n\in \mathbb{N} such that f(n)=s_{n}.

To show uniqueness, we need to demonstrate that if f(n)=f(m) then n=m. The easiest way to accomplish this is to use proof by contrapositive, that is, the fact that

f(n)=f(m)\Rightarrow n=m

is equivalent to

n\neq m\Rightarrow f(n)\neq f(m)

Let n,m\in \mathbb{N} be such that n\neq m. Without loss of generality, assume n < m (otherwise rename them). Then it follows that there exist l>0:l\in \mathbb{N} such that n=m+l. Therefore

f(n)=f(m+l)

f(m+l)=10^{-1}+10^{-2}+\cdots+10^{-m}+10^{-(m+1)}+10^{-(m+2)}+\cdots+10^{-(m+l+1)}

f(m+l)=\overset{f(m)}{\overbrace{10^{-1}+10^{-2}+\cdots+10^{-m}+10^{-(m+1)}}}+10^{-(m+2)}+\cdots+10^{-(m+l+1)}

f(m+l)=f(m)+10^{-(m+2)}+\cdots+10^{-(m+l+1)}

f(m+l)=f(m)+10^{-(m+1)}\overset{f(l-1)}{\overbrace{(10^{-1}+\dots+10^{-l})}}

f(m+l)=f(m)+10^{-(m+1)}f(l-1)

Since l>0 it follows that f(l-1)\geq10^{-1}>0 and so we have

f(n)-f(m)=f(m+l)-f(m)=10^{-(m+1)}f(l-1)>0

f(n)-f(m)\neq 0

f(n)\neq f(m)

thus proving the uniqueness. Therefore f:\mathbb{N}\rightarrow S is a bijective function and so S is countable.

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Even “More” Infinite Sets

[...] my previous post (you should read it first before continuing here), I wrote about a little bit about infinite sets [...]

November 12, 2009 08:49
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