Posts Tagged ‘theory’
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 »
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, December 17th, 2009
The Story of the Infinite Hotel
Description
A detailed tale of the Infinite Hotel. Step by step story including several pictures and an explanation of the Infinite Hotel for reference.
Overview
The Infinite Hotel is a famous math story and puzzle that was thought of by David Hilbert, a German mathematician. Sometimes the Infinite Hotel is called Hilbert’s Paradox of the Grand Hotel. It states that if one person comes into the hotel and all the rooms are full, they can all move down one room and the person can then take the first room. If k number of people come into the hotel and all the rooms are full, everyone can move down k number of rooms to make room for the people that just arrived. And, if double the amount of people that are already there are looking for rooms, everyone in room n can move to room 2n, making room for all the new arrivals in the odd-numbered rooms. This example of the Infinite Hotel can be used in certain forms of mathematical induction, and also in set theory and studies dealing with infinite numbers.
Tags: algebra, arrivals, David Hilbert, double, down, German, grand, Hilbert, hotel, induction, infinite, infinity, k, move, n!, new, numbers, paradox, room, set, space, theory
Posted in Algebra | No Comments »
Friday, November 20th, 2009
Overview of the Preimage of a Set
Description
A detailed tutorial on the preimage of a set. Step by step tutorial including several examples of the preimage of a set for reference.
Overview
The preimage of a set is defined over a function. If there is a function over A and B, then we can say that y = f(x), provided that (x, y) belongs to f. Based on this definition, x is the preimage of y under f. To find the preimage, simply look for the value of x that matches with the proper value of y in any function of ordered pairs in A and B.
Tags: a, b, belongs, coordinates, defined, definition, discrete math, f, function, image, ordered pairs, preimage, set, theory, value, x, y
Posted in Discrete Math | No Comments »
Tuesday, October 27th, 2009
An Introduction to Relations
Description
A detailed tutorial on the introduction to relations. Step by step tutorial including several examples of the introduction to relations for reference.
Overview
A relation is defined as an ordered pair. However, that is not entirely accurate. A relation could either be an ordered pair or a set of ordered pairs. A relation can be used with either one or more normal sets, or one Cartesian product set. When used with a normal set, it is a set of ordered pairs. When used with a Cartesian product, it is the power set of that set.
Tags: cartesian, coordinates, discrete math, element, ordered pair, power, product, relation, set, subset, theory
Posted in Discrete Math | No Comments »
Tuesday, October 27th, 2009
Cartesian Products in Set Theory
Description
A detailed tutorial of Cartesian products in set theory. Step by step tutorial including several examples of Cartesian products in set theory for reference.
Overview
A Cartesian product is an operation that can be performed in set theory. It is named not for the multiplication that occurs, but for the way the resulting set is written: it is written in ordered pairs, just like Cartesian coordinates. Two sets are said to be multiplied, such as A and B. Whichever set is written first in the operation has its first coordinate written with the second coordinate of the second set. This continues until all coordinates have been used at least once.
Tags: cartesian, coordinates, discrete math, element, multiplication, operation, ordered pair, product, set, subset, theory
Posted in Discrete Math | No Comments »
Tuesday, October 27th, 2009
Introduction to Minkowski Space
Description
A detailed tutorial of the application of Minkowski space. Step by step tutorial including several examples of Minkowski space for reference.
Overview
Minkowski space, sometimes referred to as Minkowski spacetime, is the setting in which Einstein’s theory of relativity was formed. Three ordinary dimensions of space are combined with a single dimension of time. This makes Minkowski space a four-dimensional manifold for representing spacetime. Minkowski space is often contrasted with Euclidean space because they are the same, except that Euclidean space has no dimension of time, and Minkowski space does.
Tags: algebra, dimension, Einstein, Euclidean, manifold, Minkowski, relativity, space, spacetime, theory, time
Posted in Algebra | No Comments »
Thursday, October 22nd, 2009
Inductive Sets in Set Theory
Description
A detailed tutorial on inductive sets in set theory. Step by step tutorial including several examples of inductive sets in set theory for reference.
Overview
An inductive set is a continuous set of natural numbers that follows a basic pattern of n + 1. This means that for all numbers in the set, that number plus the number one must also be included in the set.The set does not need to include all natural numbers – that is, the set may start at any natural number provided it is greater than or equal to one. However, the set must continue to infinity or it cannot be considered an inductive set.
Tags: -1, addition, complete, continuous, discrete math, element, equal, greater, induction, inductive, infinity, mathematical, natural, numbers, one, pattern, principle, set, subset, theory
Posted in Discrete Math | No Comments »
Friday, October 9th, 2009
Notation in Set Theory
Description
A detailed tutorial of the notation in set theory. Step by step tutorial including several examples of the notation in set theory for reference.
Overview
The notation for set theory, also called set notation or set-builder notation, is simple. It consists of a special curled bracket enclosing the elements of the set. It also includes a variable, x. When using the notation for set theory, your elements will be arranged such as {x|x = …}. You could have what x is equal to, what x in not equal to, you could say that x is less than or greater than something, or that x must be something. Whatever x is, is part of your set. If x is a natural number less than 2, then your only element is 1. Reading the set and writing the set is not difficult, but can be confusing if you don’t understand that all x stands for is all the elements of the set, and has no significance outside of that.
Tags: bracket, discrete math, elements, equals, Math, notation, set, set-builder, theory, variable, x
Posted in Discrete Math | No Comments »
Friday, October 9th, 2009
Mathematical Application of the Queueing Theory
Description
A detailed tutorial on the queueing theory. Step by step tutorial including several examples of the queueing theory for reference.
Overview
The queueing theory is the study of waiting lines – from a mathematical point of view. Because of this, it is sometimes called the waiting-line theory. It is the mathematical process of arriving at the back of the line, waiting in the line, and getting to the front of the line. We should be familiar with this – it happens every time we go out shopping. But by using the queueing theory, you will be able to tell how long you will be stuck in that line for – instead of waiting to find out! In a mathematical sense, you will be able to figure out the probability of how many people are waiting in line, and how long you will be waiting in line.
Tags: algebra, line, Math, mathematical, probability, queue, queueing, queuing, theory, time, waiting
Posted in Algebra | No Comments »
