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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 inverse image of sets. Step by step tutorial on the inverse image of sets for reference. Knowledge of the inverse image of sets is important in advanced discrete mathematics courses.

Say that you have a function f: A –> B. Then, X is a subset of A and Y is a subset of B. The image of X or the image set of X is f(X) = {y belongs to B: y = f(x) for some x belonging to X}. The inverse image of Y is defined as f^-1(Y) = {x belongs to A: f(x) belongs to Y}. The inverse image is simply a reversed form of the image. Often when asked to find the inverse image, it will help to set up a drawing of the image of the function, connecting everything where it needs to go. Then to find the inverse you simply reverse your work.

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A detailed tutorial on the preimage of a set. Step by step tutorial including several examples of the preimage of a set for reference.

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.