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The formal term used for a collection of objects. It is denoted (but other kinds of brackets can be used as well), where is a nonempty set called the index set, and is called the term of index of the family.
A family with index set is called a sequence.
The union and the intersection of a family of sets are denoted
(1) |
respectively.
If all terms belong to an additive monoid, one can consider the sum
(2) |
provided the number of nonzero terms is finite, i.e., the so-called support of the family
(3) |
is a finite set. A similar argument applies to multiplicative monoids, and to the product
(4) |
up to replacement of the zero element with the identity element 1.
According to its formal definition (Bourbaki 1970), if the terms belong to the set , the family is a map , where for all .
Every set gives rise to a family
(5) |
from which the original set can be recovered as the range of . Accordingly, every family also gives rise to a set
(6) |
from which, however, the original family in general cannot be recovered.
Bourbaki, N. Eléments de Mathématiques. Théorie des Ensembles. Paris, France: Hermann, p. ER11, 1970.
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