following are some facts related to bijections: A function is bijective if and only if it is invertible, that is, there is a function such that identity function on X and identity function on . This function maps each image to its unique preimage. The composition of two bijections is again a bijection, but if is a bijection, then it can only be concluded that is injective and is surjective (see the figure at right and the remarks above regarding injections and surjections). The bijections from a set to itself form a group under composition, called the symmetric group. Cardinality Suppose that one wants to define what it means for two sets to "have the same number of elements". One way to do this is to say that two sets "have the same number of elements", if and only if all the elements of one set can be paired with the elements of the other, in such a way that each element is paired with exactly one element. Accordingly, one can define two sets to "have the same number of elements"—if there is a bijection between them. In which case, the two sets are said to have the same cardinality. Likewise, one can say that set "has fewer than or the same number of elements" as set , if there is an injection from to ; one can also say that set "has fewer than the number of elements" in set , if there is an injection from to , but not a bijection between and . Examples It is important to specify the domain and codomain of each function, since by changing these, functions which appear to be the same may have different properties. Injective and surjective (bijective) The identity function idX for every non-empty set X, and thus specifically , and thus also its inverse The exponential function (that is, the exponential function with its codomain restricted to its image), and thus also its inverse the natural logarithm Injective and non-surjective The exponential function Non-injective and surjective Non-injective and non-surjective Properties For every function , subset of the domain and subset of the codomain, and . If is injective, then , and if is surjective, then . For every function , one can define a surjection and an injection . It follows that . This decomposition is unique up to isomorphism. Category theory In the category of sets,