<Li> A function f: X → Y is bijective if and only if it is invertible, that is, there is a function g: Y → X such that g o f = identity function on X and f o g = identity function on Y . This function maps each image to its unique preimage . </Li> <Li> The composition of two bijections is again a bijection, but if g o f is a bijection, then it can only be concluded that f is injective and g is surjective . (See the figure at right and the remarks above regarding injections and surjections .) </Li> <Li> The bijections from a set to itself form a group under composition, called the symmetric group . </Li> <P> Suppose you want 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, we can define two sets to "have the same number of elements" if there is a bijection between them . We say that the two sets have the same cardinality . </P>

Example of one to one and onto function