Labelling and Selection With or Without Repetition
When viewing ƒ as a labelling of the elements of N, the latter may be thought of as arranged in a sequence, and the labels as being successively assigned to them. A requirement that ƒ be injective means that no label can be used a second time; the result is a sequence of labels without repetition. In the absence of such a requirement, the terminology "sequences with repetition" is used, meaning that labels may be used more than once (although sequences that happen to be without repetition are also allowed).
For an unordered selection the same kind of distinction applies. If ƒ must be injective, then the selection must involve n distinct elements of X, so it is a subset of X of size n, also called an n-combination. Without the requirement, a same element of X may occur multiple times in the selection, and the result is a multiset of size n of elements from X, also called an n-multicombination or n-combination with repetition.
In these cases the requirement of a surjective ƒ means that every label is to be used at least once, respectively that every element of X be included in the selection at least once. Such a requirement is less natural to handle mathematically, and indeed the former case is more easily viewed first as a grouping of elements of N, with in addition a labelling of the groups by the elements of X.
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