FN×N

Fn×n is a mathematical concept that refers to the Cartesian product of a set with itself n times. In other words, it is the set of all possible n-tuples (ordered n-element lists) whose entries are drawn from the set Fn. The notation Fn×n is often used in the context of combinatorics, graph theory, and theoretical computer science.

The set Fn is typically defined as the set of all functions from the set {1, 2, ..., n} to itself. Therefore, Fn×n can be thought of as the set of all possible functions from an n-element set to itself. This set has a cardinality of n^n, as there are n choices for each of the n inputs.

Fn×n is closely related to the concept of permutations. In fact, the set of all permutations of an n-element set is a subset of Fn×n, specifically the subset of functions that are bijective (one-to-one and onto). This subset has a cardinality of n!, where n! denotes the factorial of n.

Fn×n is also used in the context of graph theory, where it can be used to represent the set of all possible graphs with n vertices. Each function in Fn×n corresponds to a graph where there is an edge between vertex i and vertex f(i) for each i in {1, 2, ..., n}.

In theoretical computer science, Fn×n is used to model the set of all possible computations of a deterministic Turing machine with n states. Each function in Fn×n corresponds to a computation where the machine transitions from state i to state f(i) for each i in {1, 2, ..., n}.

Overall, Fn×n is a fundamental concept in discrete mathematics with wide-ranging applications in various fields.