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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 injective and surjective functions. Step by step tutorial including several examples of injective and surjective functions for reference.

When given a function, there are two properties it can possess: it can be either injective or surjective. An injective function is a function that associates distinct arguments in one domain with distinct values in one codomain, and every unique argument produces a unique result. A surjective function is a function where the range is equal to the codomain. A surjective function is also called a surjection or said to be onto. For both cases, the function could be bijective if all elements in the codomain are mapped, which means that it would be both injective and surjective at the same time.

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

A linear transformation takes place between two vector spaces. For two vector spaces V and W, there is a map T such that T(v_1 + v_2) = T(v_1) + T(v_2) for any vectors v_1 and v_2 in V, and T(a  v) = a T(v) for any scalar a. Examples of linear transformation are often obtained through matrix multiplication. Linear transformations can also be injective or surjective