Endofunctions in Mathematics
In mathematics, an endofunction is a function whose codomain is equal to its domain. A homomorphic endofunction is an endomorphism.
Let S be an arbitrary set. Among endofunctions on S one finds permutations of S and constant functions associating to each a given . Every permutation of S has the codomain equal to its domain and is bijective and invertible. A constant function on S, if S has more than 1 element, has a codomain that is a proper subset of its domain, is not bijective (and non invertible). The function associating to each natural integer n the floor of n/2 has its codomain equal to its domain and is not invertible.
Finite endofunctions are equivalent to monogeneous digraphs, i.e. digraphs having all nodes with outdegree equal to 1, and can be easily described. For sets of size n, there are n^n endofunctions on the set.
Particular bijective endofunctions are the involutions, i.e. the functions coinciding with their inverses.
Read more about this topic: Endomorphism
Famous quotes containing the word mathematics:
“It is a monstrous thing to force a child to learn Latin or Greek or mathematics on the ground that they are an indispensable gymnastic for the mental powers. It would be monstrous even if it were true.”
—George Bernard Shaw (1856–1950)