May
20, 2017 Saturday
Bedtime
Story
Cantor's Diagonal Argument: Constructing a Unique Set s
Having
taken a brief diversion on the concept of numbers, the building blocks of
mathematics, let me take you back to Cantor and his diagonal argument.
In
his 1891 paper, Cantor considered a set T that had in it all the infinite
sequences of binary digits, meaning 1s and 0s.
Then
he begins the first part of his paper which is a constructive proof.
The
construction involves enumeration of elements of the set T as s1, s2,
s3,…,sn,…
Cantor
goes on further to elaborate each of these elements writing them down in all
possible binary forms.
For
instance,
s1
= (0, 0, 0, 0, 0, 0, 0, …)
s2
= (1, 1, 1, 1, 1, 1, 1, …)
s3
= (0, 1, 0, 1, 0, 1, 0, …)
s4
= (1, 0, 1, 0, 1, 0, 1, …)
s5
= (1, 1, 0, 1, 0, 1, 1, …)
s6
= (0, 0, 1, 1, 0, 1, 1, …)
s7
= (1, 0, 0, 0, 1, 0, 0, …)
…
This
is how it goes on.
Now
note carefully what he does.
He
takes the nth digit of each series and marks them out.
Let
me show you in the figure by emboldening each nth digit.
s1
= (0, 0, 0, 0, 0, 0, 0, …)
s2
= (1, 1, 1, 1, 1, 1, 1, …)
s3
= (0, 1, 0, 1, 0, 1, 0, …)
s4
= (1, 0, 1, 0, 1, 0, 1, …)
s5
= (1, 1, 0, 1, 0, 1, 1, …)
s6
= (0, 0, 1, 1, 0, 1, 1, …)
s7
= (1, 0, 0, 0, 1, 0, 0, …)
…
So
now if you take all these digits and form a set of digits, there is nothing new
about it.
Cantor
does something remarkable here and that is the crux of his diagonal argument.
Cantor
takes the complementary of each marked and creates a new set s.
s
= (1, 0, 1, 1, 1, 0, 1…)
Now
because the nth digit of each sn differs from that in set s, the set
s is unique from all the possible sets that can or have been listed previous to
it.
This
was the construction part of the proof.
We
shall deal the second part, that is, proof by contradiction in the nights to
come.
Till
then, keep this stuff in your memory.
Stay tuned to the voice of an average story storytelling chimpanzee
or login at http://panarrans.blogspot.in/
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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