Notes for Week #2 -- Biology S1002E (2002)

Wednesday, June 5th

We'll begin with a discussion of how to revise the syllabus. My initial proposal is that we turn it into a seminar, and integrate the mathematics into our discussions of the major texts in the history of classical genetics (rather than study the mathematics first). I'll also propose that we change the schedule so that we meet from 1-4 on both Wednesdays and Thursdays. I'm happy to revise or even drop both of these proposals based on our discussions.
We'll finish our discussion of mathematical proof, reviewing and expanding our experience with:
- Direct Proof (e.g. Gauss' sum, and the Graph Theory Theorem that says that in any graph the number of vertices with odd degree is always even).
- Indirect Proof (e.g. the irrationality of the square-root of 2,3,5 or any prime number, and the Number Theory Theorem that there is no greatest prime number).
- Mathematical Induction (e.g. the postage problem, the sum of cubes, and others from the long list in the Foundations of Math Notes.
For Next Time: Prove by induction that every polygon can be triangulated.

Thursday, June 6th

Having distinguished between mathematical and scientific methods of proof, we will look at how a mathematical theory is constructed. The theory we'll examine is elementary (it's a theory of counting!) but it serves as the foundation for sophisticated methods in a field of mathematics called Combinatorics. The fundamental theorems are:
Theorem 1: (The Fundamental Theorem of Counting): Suppose the set A1 contains n1 members, that the set A2 contains n2 members, the set A3 contains n3 members, ......., and the set A(m) contains n(M) members. Then the number of ways to choose one member from each of the m sets is equal to:

n1 x n2 x n3 x ........... x n(M)
Theorem 2: The number of permutations of n members of a set is equal to n!.
Theorem 3: The number of permutations of n members of a set arranged in r places is equal to:

Theorem 4: The number of combinations of n members of a set, chosen r at a time, is equal to:
For Next Time: Be able to explain why the following problems all have the solution C(10,5):
- How many ways can a class of 10 students be divided in half?
- How many ways are there to spell "ABRACADABRA" in Polya's famous diamond?
- How many non-descending paths are there from the point (0,0) to the point (5,5) if movement is restricted to the lines of the graph paper?
Also For Next Time:: Begin reading Mendel's "Experiments in Plant Hybridization" (available at www.esp.org and at MendelWeb)