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A detailed tutorial on the Cantor-Bernstein-Schroeder Theorem. Step by step tutorial including several examples of the Cantor-Bernstein-Schroeder Theorem for reference.

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.

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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 mathematical bridges. Step by step tutorial including several examples of mathematical bridges for reference.

The bridge is a type of mathematical structure. When an edge is taken off of a connected graph, and the resulting graph is disconnected, that edge is considered to be a bridge. Either way, the resulting graph is called a subgraph. The name “bridge” was thought up for these edges because they connect one part of the structure to another part of the structure, and are extremely important in a graph.

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A detailed tutorial on how to write step functions. Step by step tutorial including several examples of how to write step functions for reference.

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.

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A detailed tutorial on the universe of discourse. Step by step tutorial including several examples of the universe of discourse for reference.

The universe of discourse is normally just referred to simply as the universe of a set. The universe of discourse can also be the universe of a certain truth set. Basically, it is all possible considerations for a truth set. It is also a set itself, one where many different subsets are taken from. As you can tell, the universe of discourse has different meanings depending on the exact branch of math you are studying. However, all definitions have one thing in common: the universe is a set where many other sets are taken from. Normally it is easy to figure out what the universe of dicourse is based on the context of the problem you are trying to solve.

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A detailed tutorial on how to identify disconnected graphs. Step by step tutorial including several examples of disconnected graphs for reference.

A disconnected graph is a graph where not every single vertex is connected to all other vertices. Typically, graphs will have paths from all vertices, but if there is not a direct path from each and every vertex, then it is considered to be a disconnected graph. Some common shapes that are seen that are disconnected graphs are stars, rectangles, and hexagons. The opposite of a disconnected graph is a connected graph.

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A detailed tutorial on how to identify connected graphs. Step by step tutorial including several examples of connected graphs for reference.

A connected graph is a graph where every single vertex is connected to every other vertex. This does not mean to simply have a clear path from one vertex to another – it means there needs to be a direct path, or an edge, between two vertices. A triangle is a commonly seen shape that is a connected graph. The opposite of a connected graph is a disconnected graph.

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A detailed tutorial on axioms and postulates. Step by step tutorial including several examples of axioms and postulates for reference.

Axioms, sometimes called postulates, are parts of a proof. When you write a proof, there are several important parts: the given information and statements explaining the given information, the proof itself along with examples, and the conclusion. Axioms and postulates are the first set of statements that contain the given information. These statements are all assumed to be true. An important part of axioms and postulates are undefined terms, from which new concepts can be assumed, and new theorems can be deduced.

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A detailed tutorial on ordinary sets in set theory. Step by step tutorial including several examples of ordinary sets in set theory for reference.

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.