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I haven't seen a book mentioning 'an element of a family', though either 'a member of a family' or 'a member of a set' is mentioned frequently.

Indexed Families of Sets
Recall that a set is a collection of elements that are all distinct.
Roughly speaking, a family is to be considered as a collection of non necessarily distinct objects called members.
Source: Set Theory by You-Feng Lin, Shwu-Yeng T.Lin

"A set is a collection of elements that are all distinct.
A family is a collection of members which are not necessarily distinct.
Source: Set Theory Essentials by Emil G. Milewski, p.15

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Some preliminary comments: we cannot define all. Thus, we have to start from some "intuitive" concepts not defined rigorously and use them to "introduce" the undefined terms of our theory.

Those undefined terms will be the subjects of the axioms of the theory and they will be used to define rigorously new concepts of the theory.

We have to take care when we mix the two set of concepts: intuitive and rigorous ones and espacially when we use the intuitive one has "synonims" for the rigorous ones.

Having said that, if we read the initial statements of You-Feng Lin & Shwu-Yeng T.Lin's textbook, we can find [page 35]:

"What is a set?" is a very difficult question to answer. In this elementary book, we shall not go into any complicated axiomatic approach to Set Theory, but shall content ourself to accept the following: a set is any collection into a whole of definite, distinguishable objects, called elements, of our intuition or thought. This intuitive definition of a set was first given by Georg Cantor (1845-1918), who originated the theory of sets in 1895.

Here we have in play some "intuitive" concepts: collection, object.

We have to imagine a "universe" populated with objects: cats, people, numbers, collections.

This is what the authors assume, without proof nor further discussion.

Some of these objects we call them sets, like the collection $\{ John, Mary \}$ and some not, like the person $John$ or my cat.

The basic features of sets are:

(i) they have elements that are "objects" of the universe and elements of sets are "described" by the basic relation of "belongs to"*: $\in$, [unfortunately for the authors, this relation is usually called: membership relation] like in :

$John \in \{ John, Mary \}$;

and:

(ii) elements of a collection are all distinct.

Are those elements also sets ? Sometimes.

Some of the examples of sets in the book are:

The set of all chairs in this classroom.

The set of all students in this university.

The set of all natural numbers.

We may agree that chairs and students are not sets (they are not collections) and - it seems to me - that nowhere in the book it is assumed that numbers are sets.

But there are cases where we have to consider collections of sets. In this case we have a set whose elements are in turn sets, like e.g.;

$\mathcal A = \{ \{n, 2n \} \mid n \in \mathbb N \}$

where each element $A_n = \{n, 2n \}$ of the collection $\mathcal A$ is itself a collection (or set).

May we imagine "real life" examples of family of sets ? Yes: if we assume that a book is a collection (or set) of pages, then a library, being a collection of books, is a family whose elements are themselves sets.

The authors choose to call family

a collection of not necessarily distinct objects called members.

Again, we may have families whose members are not sets, like $\{ a, a, a \}$ and $\{ John, John, Mary \}$, or [see EXAMPLE 7] the family $\mathcal F$ of sets $\emptyset, \mathbb N, \mathbb Z, \mathbb Q, \mathbb R$, and $\mathbb R$.

Thus, according to the terminolgy adopted by the authors, sets have elements and families have members.

But the relation "belongs to" is symbolized in both case with: $\in$. See Def.6, page 52:

$\{ x \in U \mid x \in A \text { for some } A \in \mathcal F \}$

and thus it seems to me that there is no harm in calling a member of a family: an element of the family.


Note: the above distinction is not "standard". Compare with:

The word "family" is a common synonym for "set", and we use family of sets and set of sets interchangeably.

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  • $\begingroup$ With the quote from the Axiomatic set theory(1960), do you mean a family is different from 'a family of sets=a set of sets'? >It should be noted that a family is distinct from a set of sets, since a set may be repeated in a family, but can count only once as an element of a set. - A. G. Hamilton, Numbers, Sets and Axioms: The Apparatus of Mathematics (1982), page. 132 $\endgroup$ – buzzee May 30 '16 at 4:59
  • $\begingroup$ @buzzee - No; a family is different from a set because it allows for "repetitions". But I'm quite sure that also in your textbook, following the general definition of family, you will find only examples of families of sets. $\endgroup$ – Mauro ALLEGRANZA May 30 '16 at 7:06
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You should be aware that this distinction is by no means universally made, and I would definitely not call it standard. If it is made, the answer to your question really depends on how formal you want to be.

Formally, an indexed family $\mathscr{A}=\{a_i:i\in I\}$ is actually a function whose domain is the index set $I$, and whose codomain is some set $A$ that has each of the objects $a_i$ as an element. Thus, from a formal, technical point of view $\mathscr{A}$ is a subset of $I\times A$ with the property that for each $i\in I$ there is exactly one $a_i\in A$ such that $\langle i,a_i\rangle\in\mathscr{A}$. The elements of $\mathscr{A}$ are therefore ordered pairs of the form $\langle i,a_i\rangle$; the objects $a_i$ are not themselves elements of the family. The actual elements $\langle i,a_i\rangle$ of $\mathscr{A}$ are in fact all distinct: there’s one for each $i\in I$, and if $i,j\in I$ with $i\ne j$, then of course $\langle i,a_i\rangle\ne\langle j,a_j\rangle$ even if $a_i=a_j$. If we choose to call the objects $a_i$ members of $\mathscr{A}$, then we’re defining the word member to mean something different from the technical term element: the members of $\mathscr{A}$ in this sense are the objects $a_i$, while the elements of $\mathscr{A}$ (in the usual mathematical sense of the word represented by the notation $\in$) are the ordered pairs $\langle i,a_i\rangle$. In particular, it is technically incorrect to write $a_i\in\mathscr{A}$ unless you treat $a_i$ as an abbreviation for the ordered pair $\langle i,a_i\rangle$ rather than simply as the name of an element of $A$.

If the careful, formal distinction between member in this sense and element in the usual sense isn’t important in the context in which you’re working (and it often isn’t), then you can afford to be sloppy and speak of $a_i$ as an element of $\mathscr{A}$ and write $a_i\in\mathscr{A}$. This may even be a good idea: avoiding unnecessary technicalities often makes things easier to understand. Thus, in many contexts it can be perfectly fine to speak of an element of an indexed family; you should just be aware that you’re using element of to mean something slightly different from $\in$.

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Yes. Since "family" is a synonym for "set" and "element" is a synonym for "member". Sometimes we use the different words to assist in organizing our thoughts. For example "F is a family of subsets of S" rather than "F is a set of subsets of S". The word "collection" is also used as a synonym for "set". And we also say "x is in S" and "x belongs to S" to mean "x is a member of S".

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  • $\begingroup$ OP asks explicitly for the case where 'family' has a different meaning than 'set' $\endgroup$ – Bananach May 29 '16 at 10:08

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