Which is the correct explanation? $family ⊇ set$ or $family \Leftrightarrow$a set whose elements are sets themselves

I'm not sure about the concept of family. One book explains it as a broader concept containing set,e.g. {a, a, a}={a} another explains family as a set whose elements are sets themselves.

"Families of sets. If the elements of a set are sets themselves, then such a set is said to be 'family of sets' . The words "collection" or "class" are also used for a set of sets."
Source: Krishna's Topology, J. N. Sharma, 2014, p.6

However, in the following explanation from another book, {a, a, a}, {1, 2}, {2, 4}, {3, 6} are sets but their elements a, 1, 2, 4, 3, 6 are not sets.

Indexed Families of Sets
Recall that a set is a collection of sets that are all distinct.
Roughly speaking, a family is to be considered as a collection of non necessarily distinct objects called members. For example, {a, a, a} is a family with three members a, a, and a. But the same family {a, a, a} considered as a set is just the singleton set {a} with only one element, a. Let $\Gamma$ be a set and assume that with each element $\gamma$ of $\Gamma$ there is associated a set $A_{\gamma}$. The family of all such sets $A_{\gamma}$ is called an indexed family of sets indexed by the set Γ and is denoted by {$A_{\gamma}$|$\gamma \in \Gamma$}. For example, the family of sets: $$\{1, 2\}, \{2, 4\}, \{3, 6\} \cdots, \{n, 2n \}, \cdots$$ may be considered as an indexed family of sets indexed by the set $N$ of natural numbers, where $A_n =\{n, 2n\}$ for each $n \in N$. This family of sets may be denoted by {{2n, n} | $n\in N$}.
Source: Set Theory by You-Feng Lin, Shwu-Yeng T.Lin

Definition 8. Let X and Y be sets. A function from X to Y is a triple (f, X, Y), where f is a relation from X to Y satisfying
(a) Dom(f) = X.
(b) If (x, y)$\in f$ and (x, z) $\in f$, then y=z.
We shall adhere to the custom of writing f: $X\space \rightarrow Y$ instead of (f, X, Y) and $y=f(x)$ instead of $(x,\space y) \in f$.

Definition 9 Let $f: X\rightarrow Y$ be a function, and let $A$ and $B$ be subsets of X and Y, respectively.

(a) The image of $A$ under $f$, which we denote $f(A)$, is the set of all images $f(x)$ such that $x∈A$.
(b) The inverse image of B under f, which we denote $f^{-1}(B)$, is the set of all images of y in B.

In symbols, $f (A) =\{ f (x) \mid x\in A\}$, $f^{-1}(B)=\{x \mid f (x)\in B\}$

• I personally use "collection of sets" to mean "set of sets," and "family of sets" to mean "indexed-family of sets." But different authors adopt different conventions. – goblin May 27 '16 at 9:33
• There is no general consistency of usage. There is a tendency to use the terms family and collection for sets whose elements are themselves explicitly sets, but this is an informal distinction mostly designed to avoid excessive repetition of the word set. (If your underlying formal set theory is $\mathsf{ZF(C)}$, everything is a set anyway.) goblin’s distinction in the comment above is completely foreign to me. Formally an indexed family is a function (as drhab notes below), and informally I see no reason to distinguish indexed from non-indexed collections: after all, any ... – Brian M. Scott May 27 '16 at 16:58
• ... collection $\mathscr{A}$ can be indexed by itself. – Brian M. Scott May 27 '16 at 16:59

Personally I would advice you to think of a family $(A_{\lambda})_{\lambda\in\Lambda}$ as a function $f$ that has $\Lambda$ as its domain and has the set $\{A_{\lambda}\mid\lambda\in\Lambda\}$ as its codomain. The function is prescribed by $\lambda\mapsto A_{\lambda}$.

Note that for $\lambda_1,\lambda_2\in\Lambda$ you can have: $$\lambda_1\neq\lambda_2\wedge A_{\lambda_1}=A_{\lambda_2}$$ or equivalently: the function is not necessarily injective.

That fact gives a link to multisets. If $\Gamma\subseteq\Lambda$ then $\{f(\lambda)\mid\lambda\in\Gamma\}$ is a set, but you can also think of it as a multiset.

In that context: if $B\in\{f(\lambda)\mid\lambda\in\Gamma\}$ then the multiplicity of $B$ is the cardinality of $f^{-1}(\{B\})=\{\lambda\in\Gamma\mid A_{\lambda}=B\}$. In words the number of $\lambda$'s in $\Gamma$ with $A_\lambda=B$.

• So... you mean a family is a set whose elements are sets themselves? – buzzee May 27 '16 at 13:57
• What I mean is: a family of sets is a function from an indexset (its domain) to a collection of sets (its codomain). – drhab May 27 '16 at 17:47

I believe both examples you quoted say that families are sets whose members are sets, the difference being the first example is indexed.

For your example of sets with repeated elements like {a,a,a}, this is a multiset. See https://en.m.wikipedia.org/wiki/Multiset

• But in a family {$a, a, a$}, $a$ is not a set. – buzzee May 27 '16 at 13:59