November 24, 2016 Thursday
Bedtime Story
Defining Total Order in a set and Well Ordered Set
I will keep coming back to Cantor’s work in parallel with defining
the basic notions that arose out of his work.
While the cardinal number has something to do with the size of the
set, the concept of ordinal number arose when Cantor needed to deal with the
positions of the elements of various infinite sets.
Cantor decided to represent the cardinality of infinite sets with
the Hebrew letter aleph
.
.
So for example, take the natural numbers.
They start from 1 and go on endlessly.
Cantor took the cardinality of natural numbers to be aleph-zero or
0.
0.
This was an audacious step though it may not seem so to us ordinary
mortals.
So two sets can have the same cardinality or the same cardinal
numbers but due to their difference in the linear sequence, can have different
ordinal numbers.
Ordinal numbers deal with the ordering of elements in a set.
The difference between the cardinality and ordinality is subtle
and not obviously apparent in finite sets.
Here arises the concept of a well-ordered set and total order.
A set say A which has elements a, b and c is said to be totally
ordered if it satisfies the following conditions:
If a ≤ b and b ≤ a then a = b; possessing antisymmetry.
If a ≤ b and b ≤ c then a ≤ c; possessing transitivity.
a ≤ b or b ≤ a; possessing totality.
Thus a total ordered set is antisymmetric, transitive and is
total.
A well-order on a set A implies that the set A has total order and
if non-empty, has a least element in its ordering.
It is understandably a bit confusing, so allow me clear it with an
example which I shall do in the following bedtime story.
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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