Algorithms
Section 3.3 Complexity of Algorithms
Algorithm Complexity
Space Complexity: Determine the
approximate memory required to solve a
problem of size n.
Time Complexity: Determine the
approximate number of operations required to
solve a problem of

Basic Structures
Section 2.3 Functions
Definitions
Function: Let A and B be sets. A function (mapping,
map) f from A to B, denoted f : A B, is a subset of
AB, and is a rule that assigns to each element a A
exactly one element f(a) B, called the value of

Basic Structures
Section 2.1 Sets
Set Definition
A set is a collection of objects or elements or
members.
A set is said to contain its elements.
There must be an underlying universal set U (the set
containing everything currently under consideration),
ei

Logic
Section 1.3 Propositional Equivalences
Definitions
A tautology is a proposition which is always
true.
Classic Example: PP
A contradiction is a proposition which is
always false.
Classic Example: : PP
A contingency is a proposition which neither a

Basic Structures
Section 2.2 Set Operations
Logic and Set Theory
Propositional calculus and set theory are both
instances of an algebraic system called a
Boolean Algebra
The operators in set theory are defined in terms
of the corresponding operators in
pr

Proof
Section 5.1 Mathematical Induction
Definitions
Definition: A set S is well ordered if every subset
has a least element.
Let P(x) be a predicate over a well ordered set S.
The problem is to prove
"xP(x) .
The rule of inference called
The (first) p

Algorithms
Section 3.2 The Growth of Functions
Topics
Big-O Definition
Big-O by little-O
Complexity Classes
Properties and theorems of Big-O
Big Omega and Big Theta
The Growth of Functions
1
Overview
What really matters in comparing the
complexity o

Recursion
Section 5.3 Recursive Definitions
Recursive form
Recursive form defines a set, an equation,
or a process by defining a starting set or
value and giving a rule for continuing to build
the set, equation, or process based on
previously defined ite

Proof
Section 1.6, 7 & 8 Rules of Inference and
Intro to Proofs
Definitions
A theorem is a valid logical assertion which can be proved
using
other theorems
axioms (statements which are given to be true) and
rules of inference (logical rules which allow t

Logic
Section 1.4 & 1.5 Predicates and
Quantifiers
Predicate
A generalization of propositions - propositional
functions or predicates: propositions which contain
variables.
Examples:
Let U = Z, the integers = cfw_. . . -2, -1, 0 , 1, 2, 3, .
P(x): x > 0

Recursion
Section 5.4 Recursive Algorithms
Recursive Algorithms
A recursive algorithm is one which calls itself
to solve smaller versions of an input
problem.
Some algorithms are recursive by nature:
Binary search
Fibonacci sequence
Recursive Algorithms