Discrete Mathematics lecture notes 6-1
October 10, 2013
19. Counting with innite sets
Weve now see that we can dene the cardinality of a nite set in terms of bijections with certain special
sets of nite size, the cfw_[n]nN . In other words, we can think o
Discrete Mathematics lecture notes 12-1
November 22, 2013
31. Permutation groups, stabilizers, and orbits
Symmetric groups are all well and good, but the presence of an adjective suggests that a more general
notion is possible. In fact, weve already seen
Discrete Mathematics lecture notes 7-1
October 17, 2013
22. Algebraic and transcendental numbers
Lets solve one of the more involve homework problems in detail:
Denition 1. A number R is algebraic if there is some nonzero polynomial p(x) = qn xn + qn1 xn1
Discrete Mathematics lecture notes 10-1
November 11, 2013
27. Some applications of Inclusion-Exclusions
Last time we saw that for a collection of nite sets A1 , A2 , . . . , An , we could compute the size of the
union if we knew the size of each Ai indivi
Discrete Mathematics lecture notes 10-2
November 19, 2013
28. Introduction to the symmetric group
In this (much belated; apologies) installment, well introduce the main topic of study for the remainder
of the course. In fact, weve already met this charact
Discrete Mathematics lecture notes 11-1
November 21, 2013
29. The symmetric group is generated by transpositions
When we say a mathematical object is generated by some subset, we mean that every element element
of the object in question can be in some sen
Discrete Mathematics lecture notes 11-2
November 22, 2013
30. A combinatorial description of the parity of a permutation
In the previous lecture we proved
Theorem 1. For each n , the parity of the number of transpositions needed to express is well-dened.
Discrete Mathematics lecture notes 7-2
October 17, 2013
23. Trinomial coecients
n
i
Last time we saw that the binomial coecients
formula for powers of sums of two numbers:
n
(a + b)n =
i=0
we had introduced earlier could be used to nd a
n i ni
ab .
i
Whil
Discrete Mathematics lecture notes 6-2
October 21, 2013
20. Cantors Theorem
We close our discussion of innite cardinality by showing that, no matter how innitely big a set may
be, there is always one that is bigger.
Theorem 1 (Cantor). Let A be a set. The
Discrete Mathematics lecture notes 2-1
September 14, 2013
4. Truth as a function: Boolean algebra
After proving a handful of theorems, we start to notice that certain logical structures begin to appear
repeatedly. If we strip out the content of our argume
Discrete Mathematics lecture notes 4-1
September 24, 2013
13. Solving linear equations in Z/n
Last time we began consideration of modular number systems Z/n, and observed that, unlike in Z, a
linear1 polynomial can have multiple solutions. For example, in
Discrete Mathematics lecture notes 1-2
September 5, 2013
2. More basic number theory (as a vehicle for proving stu )
We now know what it means for the integer a to divide the integer b: There is some c such that b = a c,
and we write a|b. We used this las
Discrete Mathematics lecture notes 4-2
September 28, 2013
14. Some set theory
We have been implicitly and explicitly using sets since the beginning of this course: The natural numbers
are a set with certain properties, a well-ordered set is one in which e
Discrete Mathematics lecture notes 5-1
October 3, 2013
15. Relations
We begin with a fairly unmotivated denition:
Denition 1. If A and B are sets, a relation from A to B is a subset R A B. If (a, b) R, we write
a R b to indicate this.
Example 2. Let A = B
Discrete Mathematics lecture notes 5-2
October 3, 2013
17. Learning to count
Lets make precise an intuitive notion that weve already encountered: The size of a set.
Notation 1. For n N, let [n] be the set cfw_1, 2, . . . , n. In case that n = 0, note that
Discrete Mathematics lecture notes 12-2
November 25, 2013
32. Dividing with groups.
Lets recall two examples we saw in the previous lecture: First, is the graph
b
a
c
with group of symmetries C2 := cfw_id, (a c).1 The name C2 comes from the fact that this