November 20, 2016 Sunday
Bedtime Story
Cantor's One-to-One Correspondence
Cantor's One-to-One Correspondence
The symbol which shows that one set is a member or a subset of
another set is represented by 
.
.
The prime numbers are subsets of natural numbers which in turn are
a subset of all the integers.
The integers in their own turn are a subset of all the rational
numbers.
Then in the equation form we can represent that as:
P N Z Q
N Z Q
The symbol to show that one set is equivalent to another set is
~
Equivalence of two sets was defined (rigorously) by Cantor using
the novel concept of one-to-one correspondence of the elements of the 2 sets.
Cantor in his 1874 paper that was published in the mathematics
journal Crelle’s Journal first introduced the idea of one-to-one
correspondence.
In a later paper of 1878 yet again to the same journal, Cantor
gave a precise definition of 1-to-1 correspondence using the term “equivalence”
or the power of sets.
Two sets have the same power or equivalence if there exists a
1-to-1 correspondence between them.
Four criteria are to be fulfilled to show the existence of 1-to-1
correspondence between 2 sets X and Y.
1. Each element of X has to be paired with at least one element of
Y.
2. No element of X can be paired with more than one element of Y.
3. Each element of Y should be paired with at least one element of
X.
4. No element of Y can be pairedwith more than one element of X
The idea itself appears quite simple, even childishly simple, but
approached rigorously and taken to its logical conclusions, gave rise to
startling results.
So startling were the conclusions that Cantor came to that it
became alarming to the mathematicians who were around him that time.
It was as if all of a sudden the rug was pulled out from under
their feet when everyone felt that they were standing on firm and solid ground.
For instance the set of even integers is equivalent to the set of
odd integers.
So E ~ O.
That is pretty straightforward and very intuitive.
I shall get back to the counterintuitive part of Cantor’s work
after dealing with the basics of set theory.
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:
Some mathematical symbols are not supported by the blog. So one can get a good idea from this chart on what is missing in the text.
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