Proof by Induction
Induction is an axiom which allows us to prove
that certain properties are true for all positive
integers (or for all nonnegative integers, or all
integers >= some fixed number)
We want to highlight some basic proof techniques
using inequalities as an example.
We start from first principles and define when a
number a is greater than a number b.
Then we derive some simple facts, exercising the
The art of counting is known as enumerative
combinatorics. One tries to count the number of
elements in a set (or, typically, simultaneously
count the number of elements in a series of sets).
For example, let S1,S2,S
Propositional Logic II
Equivalences and Applications
In our formal introduction of propositional logic, we used a strict
syntax with full parenthesizing (except negations).
From now on, we will be more relaxed about the syntax
What is a Proof?
A proof is a sequence of statements, each of
which is either assumed, or follows follows from
preceding statements by a rule of inference.
We already learned many rules of inference (and
essentially all of them
Modeling with Recurrence
[From Leonardo Pisanos (a.k.a. Fibonacci) book Liber abaci]
A young pair of rabbits, one of each sex, is placed on an island.
A pair of rabbits does not breed until
Sets and Functions
Sets are the most fundamental discrete
structure on which all other discrete structures
We use naive set theory, rather than axiomatic set
theory, since this approach is more intuitive. The
Complexity of Algorithms
The sequence of Fibonacci numbers is defined
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, .
Fn1 + Fn2
Fn = 1
if n > 1
if n = 1
if n = 0
Fn1 + Fn2
Fn = 1
Recursion and Structural
Inductively Defined Sets
Consider the set
A = cfw_3,5,7,.
There is a certain ambiguity about this definition of
the set A.
Likely, A is the set of odd integers >= 3.
[However, A co
A function P from a set D to the set Prop of
propositions is called a predicate.
The set D is called the domain of P.
Let D=Z be the set of integers.
Let a predicate P: Z -> Prop be given by
P(x) = x
Sequences and Summations
A sequence is a function from a subset of the set
of integers (such as cfw_0,1,2,. or cfw_1,2,3,.) to
some set S.
We use the notation an to denote the image of
the integer n. We call an a t