Surjective function

https://en.wikipedia.org/wiki/Surjective_function

A function f from a set X to a set Y is a surjective function (also known as onto or a surjection), if for every element y in the codomain Y of f, there is at least one element x in the domain X of f such that f(x) = y. It is not required that x be unique; the function f may map one or more elements of X to the same element of Y.

dom >= cod = ran

• A mapping f: A -> B that has the existence property "for each element b of B there is an element x of A for which b = f(x)" is called a surjective or onto mapping.

• Informally, mapping is surjective if all elements in the codomain are involved in the mapping, but some elements are double-mapped. If no element in B were double-mapped, then the mapping would be injective; and since the entire codomain is involoved, the mapping would have been bijective (one-to-one and onto).

• Surjective mapping means that the codomain and the range of f are equal, cod(f) = ran(f).

• Surjective mapping means that the domain is larger then the codomain (or range), dom(f) > cod(f), since some elements in B are double-mapped.

• dom(f) : cod(f) : ran(f)

  • dom(f) > cod(f)

  • cod(f) = ran(f)

  • dom(f) > ran(f)

• Surjection is not invertable since some elements in the codomain are double mapped. Surjection irreversably collapses the domain into the range.