Random Variables, Expectations
In a previous example, we considered the situation where two dice are rolled and asked
what is the probability that the sum of the numbers is 4. We answered this by letting
E denote the event that the sum of the numbers is 4
Connectivity in Graphs
Denition 1 In a graph G, a path of length k is a sequence of k + 1 distinct vertices
v0 , v1 , . . . , vk so that cfw_vi , vi+1 is an edge in G for i = 0, . . . , k 1. A cycle of length
k is like a path of length k except its rst a
5.1 Mathematical Induction
Let P (n) be a statement regarding integer n. In many instances, the goal is to show
that P (n) is true for every positive integer n. We can of course try to do this using
the previous proof methods weve discussed e.g., direct m
A proof is a valid argument that establishes the truth of a mathematical statement. It
makes use of the hypothesis of the statement (if there are any), axioms assumed to be
true, denitions of terms, and previously proven statements. Using these
1.4 Predicates and Quantiers
Weve talked about some statements before which are not propositions. For example,
Study hard! or Whats next?. There were also statements like x > 3, x+y = z
which are almost propositions. Why almost?
Denition: A propositional
Denition: A set S is an unordered collection of distinct objects, called elements of
the set. We let a S to mean that a is an element of set S. (Reminder: the order of
the elements as well as their multiplicites do not matter!)
For example, the s
6.1 The Basics of Counting
Heres our basic goal: given a set S, determine |S|, the size of S. Many counting
problems can be solved by applying one or a combination of the principles below.
If you read them carefully, notice that most of them are not new;
Denition: A proposition is a statement that has a truth value. It is either true or
false but not both.
Which of the following statements are propositions?
1. It was very dry this summer.
2. When will this class end?
3. 32 + 42 = 62 .
4. 3x + 5 =
7.1 Discrete Probability
Denition In the context of probability, the sample space is the set of all possible
outcomes. An event is a subset of the sample space.
Example Suppose two dice are thrown. What is the sample space? What does the
event where the s
Denition: Let A and B be sets. A function from A to B, f : A B, is a rule or
an assignment that assigns each element of A to exactly one element of B.
Denition: Let f : A B be a function.
We say that A is the domain
Graphs were rst described by the famous mathematician Leonhard Euler to solve the
Knigsberg bridge problem. Since then, graphs have been used to model problems in
many areas including the social sciences, chemistry, engineering, mathematics, and
1.6 Rules of Inference
Suppose we know certain propositions to be true. What can we conclude? Heres a
logic problem from Martin Gardner:
Professors White, Brown and Black were lunching together. Isnt it remarkable,
said the lady, that our names are White,