"The Equivalence Theorem"
Saturday June 20
10AM to 11 AM
The
EQUIVALENCE THEOREM (a.k.a. Bernstein-Schröder Theorem)
Modern set theory originated between 1874 – 1884 from the researches
of Georg Cantor at Universität Halle, Germany. Around 1895, Cantor
presented the EQUIVALENCE CONJECTURE:
Given sets A and B,
if A is equivalent to a subset of B
and B equivalent to a subset of A,
then A is equivalent to B.
While this may seem obvious to many, it was not until 1898 that
Cantor’s student, Felix Bernstein proved the theorem without flaw.
In this presentation we will
review
relevant concepts of set theory: types of mappings of sets and their
behaviour when sets are combined by union, intersection and difference
(relative complement); sizes of sets and equivalence; partitions of
sets; special attention will be given to properties of chains of sets
(nested subsets).
Then we will
demonstrate two approaches for proof of the Equivalence Theorem
(the Cantor-Bernstein-Schröder Theorem): 1) à la Bernstein: 2) a
stream-lined proof for contrast based on defining a bijective function
(1-to-1 from A onto B).
Interesting examples of
applications of this theorem will be given that
give insight to distinct “DEGREES OF INFINITY”.
Relevant
background for this is basic set theory, high school-level mathematics and a normal, human logic.
The
goal of this is to help members
of the audience understand the strategy of attack in the subtle proof of
this important mathematical theorem whose statement is so deceptively
simple. Secondarily, this instruction illuminates aspects of
how one learns mathematics.
Robert A Hendrix
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