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 symbol n to denote the number of k-element subsets
k
of a
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
proving a theorem to seeing the peak of a mountain and tr
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 (so 30 people in total). If we select 16 people then
at
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
certainty, and we require that some outcome in S happen, so we
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
Universal Quantiers: - for all, for each, for every.
Existen
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 dice-throwing. . . Robert Kanigel,
The Man who Knew Inn
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 work is demonstrative reasoning,
a proof, but the proof
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 of P.
Example. Let A = cfw_1, 2, 3, 4.
P1 = cfw_1, 2, c
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
4
|S| = 36. So P (E) = 36 = 1 .
9
Notice P (cfw_(1, 4)
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, a sort of gloried dice-throwing. . . Robert Kanigel,
T
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: let S be the set of all nite sets of integers, and dene
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 logic-based public bathroom humor I know is: the dierence
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 beginning, the king said, and
stop when you get to an
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 Menschenwerk
Leopold Kronecker, quoted by Heinrich Weber.
Th
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 mathematician: those who can count, and those who
cant Anon
There a