Algorithms
(Rosen 5th ed., 2.1)
2.1: Algorithms
The foundation of computer programming.
Most generally, an algorithm just means a definite procedure for performing some
sort of task.
A computer program is simply a description of an algorithm in a language
The Theory of Sets
(Rosen 5th ed., 1.6-1.7)
Introduction to Set Theory (1.6)
A set is a new type of structure, representing an unordered collection (group,
plurality) of zero or more distinct (different) objects.
Set theory deals with operations between,
Combinatorics
1
Combinatorics
The study of the number of ways to put
things together into various combinations.
E.g. In a contest entered by 100 people,
how many different top-10 outcomes could
occur?
E.g. If a password is 6-8 letters and/or
digits, ho
Matrices
(Rosen 5th ed., 2.7)
A matrix is a rectangular array of objects (usually numbers).
An mn (m by n) matrix has exactly m horizontal rows, and n vertical columns.
Plural of matrix = matrices
An nn matrix is called a square matrix, whose order is n.
Predicate Logic (1.3)
Predicate logic is an extension of propositional logic that permits concisely
reasoning about whole classes of entities.
Propositional logic (recall) treats simple propositions (sentences) as atomic
entities.
In contrast, predicate l
Nesting of Quantifiers
Example: Let the u.d. of x & y be people.
Let L(x,y)=x likes y (a predicate w. 2 f.v.s)
Then y L(x,y) = There is someone whom x likes. (A predicate w. 1 free variable, x)
Then x (y L(x,y) =
Everyone has someone whom they like.
(A _
Probability Theory
1
Why Probability?
In the real world, we often dont know
whether a given proposition is true or false.
Probability theory gives us a way to reason
about propositions whose truth is uncertain.
Useful in weighing evidence, diagnosing
p
Basic Proof Methods
(Rosen 5th ed., 1.5 & 3.1)
Nature & Importance of Proofs
In mathematics, a proof is:
a correct (well-reasoned, logically valid) and complete (clear, detailed)
argument that rigorously & undeniably establishes the truth of a
mathematic
Foundations of Logic
(Rosen 5th ed., 1.1-1.4)
Mathematical Logic is a tool for working with complicated compound statements. It
includes:
A language for expressing them.
A concise notation for writing them.
A methodology for objectively reasoning about
Functions
(Rosen 5th ed., 1.8)
From calculus, you are familiar with the concept of a real-valued function f,
which assigns to each number xR a particular value y=f(x), where yR.
But, the notion of a function can also be naturally generalized to the concep
Propositional Equivalence (1.2)
Two syntactically (i.e., textually) different compound propositions may be the
semantically identical (i.e., have the same meaning). We call them equivalent. Learn:
Various equivalence rules or laws.
How to prove equivalenc