Archive for the ‘Math’ Category
Tuesday, December 29th, 2009
Introduction to Ordinary Sets
Description
A detailed tutorial on ordinary sets in set theory. Step by step tutorial including several examples of ordinary sets in set theory for reference.
Overview
You may be reading this and asking yourself, what is an ordinary set? An ordinary set is a set where the complete set is not part of the set. This is not the same as a subset, for as we know all sets are subsets of themselves. An example of an ordinary set is the set of all pencils. The set of pencils is not a pencil, so it is considered an ordinary set. However, the set of all thoughts is a thought. So, that set is not ordinary. In general, all sets are ordinary sets except for certain thoughts and concepts.
Tags: concepts, discrete math, element, example, extraordinary, numbers, objects, ordinary, set, subset, theory, thoughts, unusual
Posted in Discrete Math | No Comments »
Tuesday, December 29th, 2009
Overview of the Trichotomy Property
Description
A detailed tutorial on the trichotomy property. Step by step tutorial including several examples of the trichotomy property for reference.
Overview
The trichotomy property is one of the ordering properties of natural numbers. It tells us what order you need to put the natural numbers in – in other words, it tells you the placement of each element of the set of natural numbers. The trichotomy property states that is there are two natural numbers m and n, that m must be either less than n, equal to n, or greater than n. The smaller number is to be placed first, with the larger number after it. If the numbers are equal, then only one number needs to be included as part of the set.
Tags: arithmetic, element, equal, greater, inequality, larger, less, natural, number, order, placement, property, set, smaller, than, trichotomy
Posted in Arithmetic | No Comments »
Tuesday, December 29th, 2009
How to Use the Product Rule in Algebra
Description
A detailed tutorial on the algebraic product rule. Step by step tutorial including several examples of the algebraic product rule for reference.
Overview
There are many product rules in the world of math. This tutorial focuses on a product rule that is used in algebra and statistics. The product rule states that if two independent tasks T1 and T2 are to be performed, then T1 can be performed m ways and T2 can be performed n ways. Therefore, the number of ways the tasks can be performed together is m * n ways. Remember that this is only the number of possible ways to do something, not how much time it takes to do something. Also, the same method is used no matter how many different tasks you are given.
Tags: algebra, combination, multiplication, multiply, number, permutation, product, rule, statistics, task
Posted in Algebra | 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 »
Thursday, December 24th, 2009
Finding the Function of a Directed Graph
Description
A detailed tutorial on finding the function of a directed graph. Step by step tutorial including several examples of finding functions of digraphs for reference.
Overview
A directed graph, more commonly known as a digraph, is the visual representation of a function or of a relation. As in any graph, there are points and lines – called vertices and edges in a digraph. Each edge has an arrow pointing to a vertex. The first vertex – the one the arrow comes from – is the x coordinate of an ordered pair. The second vertex – the one the arrow is pointing to – is the y coordinate of an ordered pair. In the case of double-sided arrows, two ordered pairs are made, with the x and y coordinates switching. This is done for every single vertex and edge on the graph.
Tags: arrow, coordinate. ordered, digraph, directed, discrete math, double, edges, expression, First, function, graph, lines, pair, points, relation, representation, second, side, vertex, vertices, visual, x, y
Posted in Discrete Math | No Comments »
Friday, December 18th, 2009
Finding the Canonical Form of an Object
Description
A detailed tutorial of finding the canonical form of an object. Step by step tutorial including several examples of finding the canonical form of an object for reference.
Overview
Canonical form is also referred to as normal form or standard form. The canonical form of an object is a standard way of presenting that object. The process of finding a canonical form of something is referred to as canonization. Sometimes the word canonicalization is used instead. Canonical forms of objects are closly linked to differential forms of equations and numbers, and equivalence relations.
Tags: canonical, canonicalization, canonization, differential, discrete math, equation, equivalence, finding, form, normal, number, object, presenting, process, relations, standard
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 »
Friday, December 18th, 2009
An Overview of Topology
Description
A detailed tutorial on the mathematical study of topology. Step by step tutorial including several examples of topology for reference.
Overview
Topology is a study in mathematics that deals with space and spatial properties of objects. There are several different types of topology. The most common topics, called subtopics, are point-set topology, algebraic topology, geometric topology, and low dimensional topology. Topology may be a familiar sounding name to you – doubtless you have heard of a “topographical map,” used in science classes. However, the way the topographic map is created is with the study of math known as topology.
Tags: algebra, algebraic, dimensional, geometric, low, map, point, point-set, set, study, subtopic, topic, topological, topology
Posted in Algebra | No Comments »
Friday, December 18th, 2009
Your Guide to Lattice Multiplication
Description
A detailed tutorial on lattice multiplication. Step by step tutorial including several examples of lattice multiplication for reference.
Overview
Lattice multiplication is a method that is used to multiply large numbers. It uses the multiplication of smaller numbers to figure out the product of two larger numbers. Because of this, basic knowledge of times tables is required. Lattice multiplication is compromised of boxes with diagonal lines through them. Draw the diagonal line in each box from the top right corner to the bottom left corner. The top left is for your tens place (the first digit in a two digit number) and the bottom right is for your ones place (the second digit in a two digit number). The number of boxes you have depends on the number you are multiplying – for example, if you are multiplying two one-digit numbers, there is one box. If you are multiplying two 2-digit numbers, there are four boxes. The first number is across the top, and the second down the side. Where each single digit number instersects, multiply them together using the box technique. Then, using the same pattern you drew the diagonals with, mutliply the diagonals. If you have two 2-digit numbers, there will be four diagonals. Multiply together the diagonals to come up with four numbers, and the pattern you use to put them together is going from the top down and then to the right.
Tags: algebra, box, combine, diagonal, digit, double, larger, lattice, multiplication, multiply, single, small, tables, times
Posted in Algebra | No Comments »
