November 23, 2016 Wednesday
Bedtime Story
Cardinality of a Set
As Cantor worked on the open problem of representing a function by
trigonometric series he came upon the idea of transfinite ordinals.
Transfinite numbers are such that that they are certainly larger
than anything that can be defined as finite yet are not absolutely infinite.
Just think about it for a moment and see how bizarre and counter
intuitive it sounds.
This was truly a novel concept back in 1870 something which so
many mathematical geniuses had thought and were distressed about but could
never come to this conclusion.
In a way, it was almost Einsteinian moment of mathematics, similar
to suggesting that time could contract or dilate.
The only difference was that while Einstein was made a hero for
his ideas, Cantor was cursed and condemned.
How does one understand the ordinal numbers?
In a set, the natural numbers can be used do define set in at
least 2 possible ways.
A natural number can show how many elements a set contains and
then it would become a cardinal number.
Consider these two sets:
P = {2, 3, 5, 7, 11, 13}
Z = {-2, -1, 0, 1, 2, 3}
So |P| = |Z| = 6
Both these 2 sets have the same cardinality namely 6 as these two
sets satisfies the definition of one-to-one correspondence between their
respective elements.
Another word that is used to show this 1-to-1 correspondence is
bijection or existence of bijection function.
Cantor’s greatness lay in using this bijection to infinite sets
say for example the set of natural numbers.
The natural numbers 1, 2, 3, 4… are infinite.
So also are its multiples of 10 namely 10, 20, 30, 40… are infinite.
If I were to ask you which would be more, the natural numbers or
its multiples of 10, one would always go for the former.
“The natural numbers has to be more since the tens of it are a
subset of the natural numbers” would be your argument.
Cantor using the idea of bijection showed both these infinite
series could be paired using the 1-to-1 correspondence and hence their
cardinality must be the same.
Cantor went further ahead to show that similar bijections with
natural numbers could be shown for other numbers that were apparently subsets
of the natural numbers.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.in/
Good night and my fellow cousin ape.
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, may I
suggest this large collection of Kids Songs:
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