• Every x has at least one y.

A function is a process or a relation that associates each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function. If the function is called ƒ, this relation is denoted y = ƒ (x) (read ƒ of x), the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by ƒ. The symbol that is used for representing the input is the variable of the function (one often says that ƒ is a function of the variable x).

• Every element in the codomain is the output of some element in the domain.
• Every y must get hit.
• ƒ(A) = B. That is: for every y in B, there is at least one x in A such that f(x) = y, in other words  f is surjective if and only if  f(A) = B.

In mathematics, a function ƒ from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of ƒ there is at least one element x in the domain X of ƒ such that ƒ(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.

• Every element of the function’s codomain is the image of at most one element of its domain. Every x has at most one y.
• No wavy functions allowed.
•  Each y is uniquely sourced, so injective functions can be reversed.
• Whenever ƒ(x) = ƒ(y), x = y.

In mathematics, an injective function or injection or one-to-one function—not one-to-one correspondence, which is bijection (see below)—is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function’s codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence (a.k.a. bijective function), which uniquely maps all elements in both domain and codomain to each other.

• There are no unpaired elements.

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. In mathematical terms, a bijective function ƒ: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y.