Posts Tagged ‘theorem’
Tuesday, January 5th, 2010
An Overview of the Cantor-Bernstein-Schroeder Theorem
Description
A detailed tutorial on the Cantor-Bernstein-Schroeder Theorem. Step by step tutorial including several examples of the Cantor-Bernstein-Schroeder Theorem for reference.
Overview
The Cantor-Bernstein-Schroeder Theorem states that if there exist injective functions f: A –> B and g: B –> A between the sets A and B, then there exists a bijective function h: A –> B. This means that if |A| < |B| and |B| < |A|, then they are equipollent. Equipollent is a term that is similar to equal, and is denoted in the same way. However, the word equipollent means equal in cardinality, but not in any other way.
Tags: Bernstein, bijective, Cantor, cardinality, denoted, discrete math, equal, equipollent, Ernst, Felix, function, Georg, injective, Schroeder, theorem
Posted in Discrete Math | No Comments »
Thursday, December 10th, 2009
Overview of the Bounded Monotone Sequence Theorem
Description
A detailed tutorial on the bounded monotone sequence theorem. Step by step tutorial including several examples of the bounded monotone sequence theorem for reference.
Overview
The bounded monotone sequence theorem actually has several parts to it. First, you need to find out if something is bounded above or bounded below. The sequence is bounded above if there exists a real number B such that x sub n is less than or equal to B. The sequence is bounded below if there exists a real number B such that x sub n is greater than or equal to B. If something is a bounded sequence, that means it is bounded both above and below. Absolute values are also very important in determining the bounded sequence. The bounded monotone sequence theorem states that for every bounded monotone sequence x, there is a real number L such that x sub n implies L.
Tags: above, absolute, algebra, below, bounded, boundedness, equal to, greater than, implies, less than, monotone, number, real, sequence, theorem, value
Posted in Algebra | No Comments »
Thursday, November 12th, 2009
How to Identify Pythagorean Triples
Description
A detailed tutorial on Pythagorean triples. Step by step tutorial including several examples of Pythagorean triples for reference.
Overview
A Pythagorean triple is a set of three numbers that make up a right triangle. They are the measure of the sides, not the measure of the angles. This you should know by looking at the name. The Pythagorean theorem deals with only the sides of the right triangle, so Pythagorean triples should also only deal with the sides of a right triangle. All the numbers must be integers, and they must be positive. They are written rather like coordinates are, in a (a, b, c) pattern. A common example is is (3, 4, 5). From any triple, any other triple can be found. If (a, b, c) is a triple, then (ka, kb, kc) also must be a triple, according to the rule of similar triangles.
Tags: angles, Geometry, integer, measure, multiple, number, positive, pythagorean, right, sides, similar, theorem, three, triangle, triples
Posted in Geometry | No Comments »
Thursday, November 5th, 2009
Overview of the Dominated Convergence Theorem
Description
A detailed tutorial on the dominated convergence theorem. Step by step tutorial including several examples of the dominated convergence theorem for reference.
Overview
Unlike the monotone convergence theorem, the dominated convergence theorem only has one form. The official name of the theorem is Lebesgue’s Dominated Convergence Theorem, but most people just call it the dominated convergence theorem. It is considered to be a special version of the Fatou-Lebesque theorem, so Fatou’s lemma is used in direct proofs of this theorem. This theorem is also closely related to the bounded convergence theorem.
Tags: bounded proof, Calculus, convergence, direct, dominated, Fatou, form, Lebesque, lemma, monotone, special, theorem, version
Posted in Calculus | No Comments »
Thursday, November 5th, 2009
Overview of the Monotone Convergence Theorem
Description
A detailed tutorial on the monotone convergence theorem. Step by step tutorial including several examples of the monotone convergence theorem for reference.
Overview
There are several different theorems that the term “monotone convergence” can apply to. However, the most important one, and the one most common called the monotone convergence theorem, is the Lebesgue Monotone Convergence Theorem. This particular monotone convergence theorem deals with calculus, and with integrals and limits specifically. It is a more general form of the other two monotone convergence theorems, which is why it is considered to be the most important.
Tags: Calculus, converge, convergence, form, general, integral, Lebesgue, limit, monotone, number, real, sequence, series, theorem
Posted in Calculus | No Comments »
Thursday, November 5th, 2009
Introduction to Nested Intervals
Description
A detailed tutorial on nested intervals and the nested interval theorem. Step by step tutorial including several examples of nested intervals for reference.
Overview
Nested intervals means to have one interval (or multiple intervals) inside of another interval. The intervals will get smaller and smaller the more you add, until they will finally dimish entirely. There is a theorem for nested intervals, called the nested interval theorem. It states that if A_n = [a_n, b_n] is a sequence of closed intervals such that A_n+1 is a subset of A_n for all n belonging to the set of natural numbers, then the union over A_n is not an empty set.
Tags: algebra, closed, empty, interval, natural, nested, number, open, sequence, set, subset, theorem
Posted in Algebra | No Comments »
Friday, October 23rd, 2009
How to Find the Length of a Vector
Description
A detailed tutorial on finding the length of a vector. Step by step tutorial including several examples of how to find the length of a vector for reference.
Overview
The length of a vector is also known as the magnitude of a vector. This can be compared to the absolute value of a real number. In order to find the length of a vector, you need to use the Euclidean norm:
The Euclidean norm is a consequence of the Pythagorean theorem.
Tags: absolute value, algebra, consequence, Euclidean, length, magnitude, norm, pythagorean, theorem, vector
Posted in Algebra | No Comments »
Friday, October 16th, 2009
Overview of the Conjugate Zeros Theorem
Description
A detailed tutorial on the conjugate zeros theorem. Step by step tutorial including several examples of the conjugate zeros theorem for reference.
Overview
The conjugate zeros theorem states that if a + b * i is a zero of a polynomial with real coefficients, then so is a – b * i. The conjugate zeros theorem can be proved by taking any function in this form and setting it equal to zero. The conjugate zeros theorem makes many equations easier to solve, especially complex equations when you get to higher levels of math.
Tags: a, b, Calculus, complex, conjugate, equations, function, i, imaginary, Math, number, theorem, zero, zeros
Posted in Calculus | No Comments »
Tuesday, October 6th, 2009
Definition of a Uniqueness Theorem
Description
A detailed tutorial on the uniqeness theorem. Step by step tutorial including several examples of how to solve a uniqueness theorem for reference.
Overview
A uniqueness theorem is any mathematical theorem that states only one mathematical object satisifies special conditions – that is, that a problem only has one solution, a unique solution. Sometimes the solution to the equation is also determined uniquely – that is, there is only way to solve the problem, instead of multiple ways.
Tags: arithmetic, condition, existence, Math, solution, solve, theorem, unique, uniqueness
Posted in Arithmetic | No Comments »
Thursday, October 1st, 2009
How to Solve Logarithms Using the Change-of-Base Rule
Description
A detailed tutorial on solving logarithms with the change-of-base rule. Step by step tutorial including several examples of how to solve logarithms using the change-of-base rule for reference.
Overview
The change-of-base rule is typically only used when solving logarithms with a calculator. It allows you to use a number besides the calculator presets. Tha change-of-base rule states that:
In this formula, b must not be equal to one, as the logarithm of one is simply zero. This formula also implies that all logarithms are similar to each other.
Tags: algebra, base, calculator, change, change-of-base, log, logarithm, Math, rule, similar, theorem
Posted in Algebra | No Comments »
