Understanding Indexed Sets: Exploring the Concept and Notation

[bonfire09]

I barely started out learning on my own about proofs from this book called A transition to advanced math 2nd edition by chartrand. I am having trouble understanding what an indexed set is and the notation. Is there any online resources I can use to help me understand this concept?

 

[Number Nine]

What is it about indexed sets that you don't understand? All that's happening is that we have a collection of sets, and we label each of them using a subset of the natural numbers. If we have an indexed family of sets A_i with i = {1, 2, 3, 4, 5} then we have five sets labeled A₁, A₂, A₃, etc.

 

[Stephen Tashi]

we label each of them using a subset of the natural numbers.

He might be dealing with advanced mathematics, where the index set could be the rational numbers, or the real numbers or any sort of set.

bonfire09,

I suggest you post some specific questions here.

Before that, I suggest that you get familiar with the way that LaTex behaves on this forum. See https://www.physicsforums.com/showthread.php?t=546968, if you aren't already familiar with it. It woulld be a good long-term investment if you keep studying math.

 

[bonfire09]

yeah I am talking about advanced mathematics. I am using this book to help me transition better into advanced math like intro to real analysis and abstract algebra.

 

[mbs]

Indexed set notation is used to represents a one-to-one function relating the members of the indexing set to members of the indexed set. Usually the indexing set is the natural numbers, but it could be any set, including the set of real numbers. Often the indexed set is a collection of sets, i.e. a set who's members are all sets of a certain type.

A mathematical statement stating the existence of a hypothetical collection of sets indexed by any set (of arbitrarily large size) is valid as long as one accepts the Axiom of Choice. The Axiom of Choice and the notion of arbitrarily large indexed sets was somewhat controversial before the latter part of the 20th century. Rather recently AOC was shown to be consistent and independent relative to the other commonly used axioms of set theory. Now it's generally accepted and considered a necessary tool for a decent chunk of modern mathematics.

 

[SteveL27]

yeah I am talking about advanced mathematics. I am using this book to help me transition better into advanced math like intro to real analysis and abstract algebra.

Suppose you have a set, call it A.

Suppose you have another set. Ok, call it B.

Now suppose you have a LOT of sets. You can call them A, B, C, ...

Or you could call them A₁, A₂, A₃, ...

The latter notation using numeric subscripts is more convenient when you are discussing a lot of sets at the same time.

So let's say we have a countably infinite collection of sets, A₁, A₂, A₃, ...

We can think of the subscripting as a function from the natural number to some large collection of sets. In other words we have a function that for each number n, gives you the set A_n. That's how we think of an infinite collection of indexed sets.

Now that we've noticed that a collection of indexed sets can be thought as a function from an indexing set to some other collection of sets; then there's no reason the indexing set can't be more general than the natural numbers.

What if for every real number $\alpha$, we had some set $A_{\alpha}$. Then we'd have an uncountable collection of sets, indexed by the reals.

Can we think of an example of something like that? Sure. Just let $A_{\alpha}$ be the set of reals less than or equal to $\alpha$.

So $A_{6}$ is the set of reals less than or equal to 6; $A_{\pi}$ is the set of all reals less than or equal to $\pi$.

All this is just a conceptual framework for an idea that's already familiar. If we had a lot of sets to keep track of, we'd assign each one a number. The assignment can be thought of as a function from an index set to some collection of sets. The index set is commonly the natural numbers; but in general the index set could be any set. That's everything you need to know about indexed sets.