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Let and denote sets, and let

be a mapping. Then is called injective, if for two different elements , also and

are different.

If we want to show that a certain mapping is injective then we may show the following: For any two elements and fulfilling the condition , we can deduce that . This is often easier to show than the statement that implies .

Let and denote sets, and let

be a mapping. Then is called surjective, if for every , there exists at least one element , such that

We consider a football game as the mapping which assigns, to every goal of team , the corresponding goal scorer. Suppose that there are no own goals and no changes. The goals of are numbered by . Then we have a mapping

given by

The injectivity of means that every player has scored at most one goal, the surjectivity means that every player has scored at least one goal.

Let denote the set of all (living or dead) people. We study the mapping

which assigns to every person his or her (biological) mother. This is well-defined, as every person has a uniquely determined mother. This mapping is not injective, since there exists different people (brothers and sisters) with the same mother. It is also not surjective, since not every person is a mother of somebody.

The mapping

is neither injective nor surjective. It is not injective, because the different numbers and are both sent to . It is not surjective, because only nonnegative elements are in the image (a negative number does not have a real square root). The mapping

is injective, but not surjective. The injectivity can be seen as follows: If , then one number is larger, say

But then also , and in particular . The mapping

is not injective, but surjective, since every nonnegative real number has a square root. The mapping

is injective and surjective.

Let and denote sets and suppose that

is a mapping. Then is called bijective if is injective as well as


The question, whether a mapping has the properties of being injective or surjective, can be understood looking at the equation

(in the two variables and ). The surjectivity means that for every there exists at least one solution

for this equation, the injectivity means that for every there exist at most one solution for this equation, and the bijectivity means that for every there exists exactly one solution for this equation. Hence surjectivity means the existence of solutions, injectivity means the uniqueness of solutions. Both questions are everywhere in mathematics and they also can be interpreted as surjectivity or injectivity of suitable mappings.

Let denote a bijective mapping. Then the mapping

which sends every element to the uniquely determined element with ,

is called the inverse mapping of .

Let and denote sets, let


be mappings. Then the mapping

is called the composition of the mappings

and .

So we have

where the left hand side is defined by the right hand side. If both mappings are given by functional expressions, then the composition is realized by plugging in the first term into the variable of the second term (and to simplify the expression if possible).

Let and be sets and let


be mappings. Then