16. Binomial Coefficients
Question: How many subsets does a set with n elements have? 2n
Question: How many subsets with k elements does a set with n elements have?
Denition. Let n, k N. We use the sy
These are brief notes for Section 1.3: they are not complete, but they are a guide to what I want to say
today.
When asked what it was like to set about proving something, the mathematician likened
pr
24. The Pigeonhole Principle
Example: There are 22 seats in this room. If there were 23 students in the class, then at
least 2 people would have to share a seat.
Example: Suppose there are 15 couples
29. Sample Space
We have a set S of possible outcomes, and for each element s S we have a probability
P (s) of that event happening. We say that something happens with probability 1 if it is a
certain
You may fool all the people some of the time; you can even fool some of
the people all the time; but you cant fool all of the people all the time.
attributed to Abraham Lincoln.
10. Quantifiers
Unive
These are brief notes for the lecture on 3.13: they are not complete, but they are a guide to
what I want to say today. They are not guaranteed to be correct.
. . . . combinatorics, a sort of gloried
These are brief notes for the lecture on 1.4: they are not complete, but they are a guide
to what I want to say today. They are not guaranteed to be correct.
The result of the mathematicians creative
15. Partitions
Denition. Let A be a non-empty set. A partition P (or sometimes a set partition) of A
is a set of pairwise disjoint, non-empty sets whose union is A. The sets in P are called the
parts
30. Events
Example: (Pair of dice) Two dice are tossed. S = cfw_1, 2, 3, 4, 5, 6 cfw_1, 2, 3, 4, 5, 6.
What is the probability we roll a sum of 5? E = cfw_(1, 4), (2, 3), (3, 2), (4, 1) So |E| = 4 and
These are brief notes for the lecture on Monday February 8, 2009: they are not complete,
but they are a guide to what I want to say today. They are not guaranteed to be correct.
. . . . combinatorics,
All things being equal, everything is trivial Anon.
14. Equivalence Relations
Recall:
Denition. An equivalence relation on a set A is a relation on A which is reexive, symmetric, transitive
Example: l
These are brief notes for the lecture on Wednesday January 20, 2009: they are not complete, but they are a guide to what I want to say today. They are not guaranteed to be
correct.
The only bit of lo
These are brief notes for the lecture on 1.5: they are not complete, but they are a guide
to what I want to say today. They are not guaranteed to be correct.
Where shall I begin he asked. Begin at the
These are brief notes for the class on Sections 1.1-1.2.
1. Joy
Read this section on your own, and do the exercise.
2. Definitions
Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Mensc
These are brief notes for the lecture on Section 2.11: they are not complete, but they are a
guide to what I want to say today. They are not guaranteed to be correct.
There are 3 types of mathematicia