In Halmos'book on naive set theory, Halmos introduced the idea of a "family of sets". He explicitly defined a family to be a function from some indexing set to an indexed set. However, whenever he talks about a family, what he seems to always be refering to is actually the range of the family. I get so confused whenever he speaks of families in his text, whether he is speaking about the function or the range of the function is very ambiguous. Does "family" then formally mean the indexing function itself, or perhaps the range of the function? Thank you in advance.
Is a Family a Function, or the Range of a Function
elementary-set-theory
Best Answer
Formally, a family is the precisely the same thing as a function. Halmos says this himself on the first page of section 9. (Note that many people would call what Halmos calls a family an "indexed family".)
However, when people use the term "family" instead of "function", it's because they want to emphasise the range of the function as being especially important for their purposes; the domain, or index set, becomes an auxiliary object. Accordingly, it is very common to abuse language and use the term family to refer to the range of a function. This abuse of language also reflects the informal way in which people think of a family $\{x_i\}_{i\in I}$ in $X$, not as a function $x:I\to X$, but as the same thing as a set, except that all of the elements have been "tagged" with a corresponding index (an element may appear multiple times, with different tags).
For example, the index set could convey a notion of "ordering". If $A$, $B$, $C$, and $D$ are sets, then we could consider the family $x:\{0,1,2,3,4,5\}\to \{A,B,C,D\}$ given by $x(0)=B,x(1)=C,x(2)=C,x(3)=A,x(4)=C,x(5)=A$. Then, in practice, we think of $x$ as being the set $\{A,B,C\}$ arranged in a "list" $B,C,C,A,C,A$. We can distinguish between elements that appear more than once in the list, since (for example) the first $A$ that appears has the index $3$, whereas the second $A$ that appears has the index $5$.