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.
