Functions, Sets and Families
Surjective functions and families are formally equivalent, as any function f with domain I induces a family (f(i))i∈I. In practice, however, a family is viewed as a collection, not as a function: being an element of a family is equivalent with being in the range of the corresponding function. A family contains any element exactly once, if and only if the corresponding function is injective.
Like a set, a family is a container and any set X gives rise to a family (xx)x∈X. Thus any set naturally becomes a family. For any family (Ai)i∈I there is the set of all elements {Ai | i∈I}, but this does not carry any information on multiple containment or the structure given by I. Hence, by using a set instead of the family, some information might be lost.
Read more about this topic: Indexed Family
Famous quotes containing the words sets and/or families:
“It is mediocrity which makes laws and sets mantraps and spring-guns in the realm of free song, saying thus far shalt thou go and no further.”
—James Russell Lowell (1819–91)
“Accidents will occur in the best regulated families; and in families not regulated by that pervading influence which sanctifies while it enhances the—a—I would say, in short, by the influence of Woman, in the lofty character of Wife, they may be expected with confidence, and must be borne with philosophy.”
—Charles Dickens (1812–1870)