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# Functions

To define a mathematical object like a function, we must first understand what a set is.

A set is an unordered collection of objects like S = {-4, 4, -3, 3, -2, 2, -1, 1, 0}. If a set S is not infinite, we use |S| to denote the number of elements, which is known as the Cardinality of the set. If A and B are finite sets, then |A&rarrb;B|=|A|&rarrb;|B|, which is known as the Cartesian product.

For each input element in a set A, a function assigns a single output element from another set B. A is called the domain of the function, and B, the codomain. A function is a set of pairs (x, y), with none of these pairs having the same first element.

Example: The function with domain {1, 2, 3, . . .}, which doubles its input is the set {(1,2),(2,4),(3,6),(4,8),...} ...

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