Proof by Induction
Andreas Klappenecker
1
Motivation
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)
2
Induction Principle
Le
A<B
Andreas Klappenecker
Motivation
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
d
Counting
Andreas Klappenecker
Counting
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
Andreas Klappenecker
Remarks
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
Proofs
Andreas Klappenecker
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
Recurrence Relations
Andreas Klappenecker
Modeling with Recurrence
Relations
Rabbits (1/3)
[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
Andreas Klappenecker
1
Sets
Sets are the most fundamental discrete
structure on which all other discrete structures
are built.
We use naive set theory, rather than axiomatic set
theory, since this approach is more intuitive. The
drawbac
Complexity of Algorithms
Andreas Klappenecker
Example
Fibonacci
The sequence of Fibonacci numbers is defined
as
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, .
Fn1 + Fn2
Fn = 1
0
if n > 1
if n = 1
if n = 0
Fibonacci Implementation
def fib(n)
Fn1 + Fn2
Fn = 1
0
return
Recursion and Structural
Induction
Andreas Klappenecker
Inductively Defined Sets
Motivating Example
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
Predicate Logic
Andreas Klappenecker
Predicates
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.
Example
Let D=Z be the set of integers.
Let a predicate P: Z -> Prop be given by
P(x) = x
Sequences and Summations
Andreas Klappenecker
Sequences
Sequences
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