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A detailed tutorial on how to determine Dedekind cuts. Step by step tutorial including several examples of Dedekind cuts for reference.

A Dedekind cut is a partition of rational numbers into two non-empty sets A and B, such that all elements of A are less than elements of B, and A has no greatest element. The cut itself is a gap that is located between A and B, which is normally found by creating a new, irrational number, and setting it in the gap. What irrational number you use depends on what numbers you have partitioned into the two sets. It is like the number line of advanced algebra, that has both rational and irrational numbers on it instead of just integers. The Dedekind cut was named after Richard Dedekind.

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A detailed tutorial on isomorphism. Step by step tutorial including several examples of isomorphism for reference.

Isomorphism is a topic and concept that is commonly used in abstract algebra. Let (G, o) and (H, *) be groups. A homomorphism h: (G, o) –> (H, *) that is one-to-one and onto H is called an isomorphism. If h is an isomorphism, we say that (G, o) and (H, *) are isomorphic. Homomorphism is the inverse of isomorphism.

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A detailed tutorial on homomorphism. Step by step tutorial including several examples of homomorphism for reference.

Homomorphism is a topic and concept that is commonly used in abstract algebra. Let (G, o) and (H, *) be groups. An mapping of h: (G, o) –> (H, *) is called a homomorphism from (G, o) to (H, *). The range of h is called the homomorphic image of (G, o) under h. Isomorphism is the inverse of homomorphism.

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A detailed tutorial on the algebraic product rule. Step by step tutorial including several examples of the algebraic product rule for reference.

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.

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A detailed tutorial on the definition of cardinal numbers. Step by step tutorial including several examples of how to define cardinal numbers for reference.

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.

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A detailed tutorial on the pigeon-hole principle. Step by step tutorial including several examples of the pigeon-hole principle for reference.

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.

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A detailed tutorial on the mathematical study of topology. Step by step tutorial including several examples of topology for reference.

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.

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A detailed tutorial on lattice multiplication. Step by step tutorial including several examples of lattice multiplication for reference.

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.

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A detailed tale of the Infinite Hotel. Step by step story including several pictures and an explanation of the Infinite Hotel for reference.

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.

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A detailed tutorial on how to join tables and charts. Step by step tutorial including several examples on how to join tables and charts for reference.

A table, also referred to as a chart, is a way to record certain information so you can match it up quickly. They are very useful and are used in business all the time. It is possible to join certain tables. Provided that the tables share at least one common element, it is possible to combine them to form a new chart. Typically when you join tables you will either increase your columns and decrease your rows, or increase your rows and decrease your columns, depending on what way your graph is oriented and what elements are the same. Sometimes rows or columns may remain the same, but if both remain the same, then that means there is no join – it means you have the same exact chart.