Difference between indexed sets

Question

Dummit and Foote's Abstract Algebra (3rd edition) defines index sets as the following:

Are all indexed sets equal? Taking $\mathbb{N}$ as the index set, without loss of generality, as $A_i = A$, is $A_1=A_2=A_3...$?

Answer 1

The author was giving an example of an indexed family of sets in which all sets in the family are equal. Not all indexed families of sets need be like this.

For example, for each $n \in \mathbb{N}$, define $$A_n = \{ 1, 2, \dots, n \}$$ Then $\{ A_n \mid n \in \mathbb{N} \}$ is a family of sets indexed by $\mathbb{N}$ such that $A_m \ne A_n$ whenever $m \ne n$.

Answer 2

If that were so it'd be a rather useless concept.

The author is giving an example of how we can have an indexed set of sets in the first place. Frankly I think its a confusing example.

A better example would be let $ABRACADBRA5 = \{5 + 2*p|p\text{ is prime}\}$. And let $ABRACADBRA37 = \{37+ 2*p|p\text{ is prime}\}$ and then I can wave my hands and say "We can define $ABRACADBRA[\text{ insert number here}] = \{\text{ insert number here} + 2*p|p\text{ is prime}\}$.

Okay that "insert number here" business is simply indexing.

Let $A_i = \{i+ 2*p|p\text{ is prime}\}$ would have been a much simpler way of putting it.

So $\{A_i\}_{i\in \mathbb N}$ is simply a set of sets. It is the set:

$\{A_i\}= \{A_1, A_2, A_3,.... \} \{\{5,6,11,15,...\},\{6,7,12,16,...\},\{7,8,13,17,....\},...... \}$.

It is a set of sets where the sets themselves are indexed by that natural numbers.

Now here's the kicker. The indexing set can be any set, even an uncountable.

For instance let $p \in \mathbb R^2$ be an point in the plane. And let $B_p = \{$ all possible lines that contain the point $p$$\}$. The $\{B_p\}_{p \in \mathbb R^2}$ is the set of all sets of lines that go through a point.

This is an indexed set of sets. And the indexing set is the uncountable $\mathbb R^2$.

....

I guess that is the author's point. Any set such as $I = \{$ fictional elephants $\}$ can be an indexing set. But rather than saying $A_{babar} = $ some set, and $A_{tantor} = $ another set, we can say $A_{babar} = A_{tantor}=A_{hathi} = ... =A_{any\ elephant} = A$. Then $\{A_e|A_e = A\forall e=\{\text{fictional elephants}\}\}$ is an indexed set and the set of all fictional elephants is the index set.