What Is Function?

This section describes what is function - Two sets of elements, X and Y, and a rule, f, which assigns to each element in X, to one element in Y.

Function - Two sets of elements, X and Y, and a rule, f, which assigns to each element in X, to one element in Y. A function is denoted as f: X -> Y.

Domain - The set of elements, X, in the above definition.

Codomain - The set of elements, Y, in the above definition.

If x is element in X, and y is the element in Y assigned to x by a function f, y is called the image of x, and denoted as y = f(x).

If y = f(x), x is called a preimage of y.

Image of Function - The set of all elements in Y which has at least one preimage in X.

1-1 Function - A function that satisfies that "if x <> y, then f(x) <> f(y)". No two elements in the domain will have the same image for a 1-1 function.

Bijection - A 1-1 function that satisfies that "image of f is Y".

Inverses Function - The function g of a bijection f that satisfies that "if y = f(x), then x = g(y)".

One-way Function - A bijection that satisfies that "the computation of its inverse function is very hard, and close to infeasible".

Trapdoor One-way Function - A one-way function that satisfies that "the computation of its inverse function becomes feasible, if additional information is given".

Permutation - A bijection that satisfies "f: X -> X".

Involution - A permutation that satisfies that "inverse function of f is f".