Posts Tagged ‘finite’
Thursday, December 31st, 2009
How to Write Step Functions
Description
A detailed tutorial on how to write step functions. Step by step tutorial including several examples of how to write step functions for reference.
Overview
A step function, also called a staircase function, is a finite linear combination composed of several different intervals. They are considered to be a piecewise constant function. The graph of a step function is often expressed as steps, or a staircase, which is how it got its name. It simply looks like several disconnected lines, with alternate open and closed ends so that it easily passes the vertical line test for functions.
Tags: closed, combination, constant, diconnected, discrete math, ends, finite, function, graph, intervals, line, linear, lines, open, piecewise, staircase, step, test, vertical
Posted in Discrete Math | No Comments »
Tuesday, December 29th, 2009
How to Construct a Cayley Table
Description
A detailed tutorial on how to construct a Cayley table. Step by step tutorial including several examples of how to construct a Cayley table for reference.
Overview
A Cayley table is a table that expresses the structure of a finite set. A Cayley table is set up by having the elements of the set across the first row, and numbers going in a numerical order of n + 1 starting at 1 down the first column. Sometimes the table is simply different ways the elements can be ordered. Other times is is a true table, where an operation is performed between two numbers in the space where they cross each other. However, a true Cayley table must be constructed using an identity skeleton. Once an identity skeleton for the finite set has been decided on, the Cayley table can be filled out using the identity skeleton. Since there is more than one possible identity skeleton for a finite set, you may have to go through a trial and error process until you find the right one.
Tags: addition, Cayley, chart, column, cross, discrete math, division, elements, error, finite, graph, identity, multiplication, operation, order, process, row, set, skeleton, subtraction, table, trial
Posted in Discrete Math | No Comments »
Friday, December 18th, 2009
How to Define Cardinal Numbers
Description
A detailed tutorial on the definition of cardinal numbers. Step by step tutorial including several examples of how to define cardinal numbers for reference.
Overview
Cardinal numbers are natural numbers that are used to measure cardinality of sets. Cardinality is a fancy way of saying the size of a set. This means the cardinality is the number of elements in a set, provided that the set is finite. If the set is infinite, something called a transfinite cardinal number is used to describe the cardinality of the set. Cardinal numbers are a very important part of set theory, even though they are not studied often or used constantly.
Tags: abstract, algebra, analysis, cardinal, cardinality, combinatorics, elements, finite, infinite, mathematical, measure, natural, number, set, set theory, size, transfinite
Posted in Algebra | No Comments »
Friday, December 18th, 2009
Explanation of the Pigeon-Hole Principle
Description
A detailed tutorial on the pigeon-hole principle. Step by step tutorial including several examples of the pigeon-hole principle for reference.
Overview
The pigeon-hole principle is an important principle in math that states that if n items are to be put into m pigeon-holes, and n > m, then at least one pigeon-hole must contain more than one item. It is thought of as an extension of the counting principle. The pigeon-hole principle was first referred to as the drawer principle, or the shelf principle. Because of this, it is commonly called Dirichlet’s box principle or Dirichlet’s drawer principle. It is most commonly used with finite sets of elements; however, this principle can also be used with infinite sets.
Tags: algebra, box, counting, Dirichlet, drawer, elements, extension, finite, infinite, leftover, more, pigeon-hole, principle, remainder, sets, shelf, theory
Posted in Algebra | No Comments »
Thursday, October 15th, 2009
Introduction to Infinite Sets
Description
A detailed tutorial on infinite sets. Step by step tutorial including several examples of infinite sets and how to identify them for reference.
Overview
There are two types of sets, finite sets and infinite sets. The tutorial will focus on infinite sets. An infinite set is a set that has at least one endpoint of infinity, which can be implied either by having infinity in the set or by having a trailing end of the set, with no number at the end. Infinite sets can either be countable or uncountable – meaning they either have a pattern you can use to follow to infinity, or there is no pattern present.
Tags: algebra, countable, element, endpoint, finite, infinite, infinity, Math, number, set, trailing, uncountable
Posted in Algebra | No Comments »
Tuesday, October 6th, 2009
Introduction to the Gram-Schmidt Process
Description
A detailed tutorial on the Gram-Schmidt process. Step by step tutorial including a visual example of the Gram-Schmidt process for reference.
Overview
The Gram-Schmidt process is a process used for orthogonalizing a set of vectors in an inner product space. What the Gram-Schmidt process does is it takes a finite and linearly independent set and converts it to an orthogonal set that spans the same amount of space.
Tags: differential equations, Erhard Schmidt, Euclidian, finite, gram-schmidt, inner product space, Jorgen Pedersen Gram, linear algebra, linearly dependent, Math, orthogonal, orthogonalizing, process, set, vector
Posted in Differential Equations | No Comments »
Tuesday, October 6th, 2009
Definition of a Finite Set
Description
A detailed tutorial on the definition of a finite set. Step by step tutorial including several examples of finite sets for reference.
Overview
There are many different types of sets, but one of the most common ones to run into a finite sets. A finite set is a set that has a finite number of elements – meaning a set with a definite number of elements, such as five, or ten. The number of elements in the set must be a natural number, and it is called the cardinality of a set. An empty set is considered to be finite, with a cardiality of zero, even though zero is not considered to be a natural number.
Tags: algebra, cardinality, element, elements, empty set, finite, infinite, Math, natural number, set, sets, zero
Posted in Algebra | No Comments »
Thursday, September 24th, 2009
Algorithms Explained
Description
A detailed tutorial on the solving of algorithms. Step by step tutorial including several example problems of how to solve algorithms for reference.
Overview
An algorithm is something that you will find at almost any level of math – the more advanced the level of math, the more advanced the algorithm will be. When you use an algorithm, what you are doing is solving a problem by using a finite sequence of instructions. The visual representation of an algorithm is a flow chart…every time you use a flow chart or sequence to solve a problem, even one that isn’t mathematical, you are using an algorithm.
Tags: algorithm, algorithms, calculation, Calculus, data processing, finite, flow charts, instructions, Math, representation, sequence, visual
Posted in Calculus | No Comments »
