Indexed Sets (Or How to Index Sets)

Date: September 30 2022

Summary: An attempt at understanding what are indexed sets and the various forms index sets can take

Keywords: #set #theory #index #infinite #finite #interval #archive

Bibliography

Indexed Sets, (Nov. 27, 2020). Accessed: Oct. 01, 2022. [Online Video]. Available: https://www.youtube.com/watch?v=ZoR93jR9Ok0

1. Reading Motivation
2. Building an Intuition for Index Sets
How To Cite
References:
Discussion:

Reading Motivation

I was very confused about nomenclature concerning indexed sets and how to interpret them.

Building an Intuition for Index Sets

An index set $I$ can be nearly anything. The following examples build intuition to that statement:

Union Example Where $I$ Is a Finite Set of Integers

For example, to build an initial intuition on what an index set actually is, we can imagine it as a set of integers. For the following problem, let's find the union of three different sets using the index set $I$ :

Let the following hold:

I = \{1, 2, 3\}

A_{1} = \{0, 1, 2, 3, 4, 5\}

A_{2} = \{-3, -2, -1, 0, 1, 2\}

A_{3} = \{-4, -2, 0, 2, 4\}

The intersection of these three sets can be denoted as follows: $\bigcup_{i \in I} A_{i}$ . This syntax is also synonymous with the following syntax that may be more easily read: $\bigcup_{i=1}^{3} A_{i}$ . Either syntax means loosely, "create the union between sets $A_{1}$ through $A_{n}$ ." The solution to this problem would be:

\bigcup_{i \in I} A_{i} = \bigcup_{i=1}^{3} A_{i} = \{-4, -3, -2, -1, 0, 1, 2,3, 4, 5\}

NOTE: $A_{1}, A_{2}, A_{3}$ are independent sets and do not form a family of sets.

Union Example Where $I$ Is a Countably Infinite Set of Integers

For example, one can also have an index set be infinite! 😱 For the following problem, let's find the union of three different sets using the index set $I$ :

Let the following hold:

I = \N = \{1, 2, 3, \dots \}

A_{i} = \{-i, 0, i\}

In this case, $A_{i}$ is a condition that prescribes the formation of independent sets. So from $A_{i}$ we could get the following sets using the infinite index set, $I$ :

: $A*{i} = {-i, 0, i} = \begin{cases} A*{1} = {-1, 0, 1}, i = 1 \ A*{2} = {-2, 0, 2}, i = 2 \ A*{3} = {-3, 0, 3}, i = 3 \ \phantom{––} \dots \end{cases} :$

The solution to this is as follows:

\bigcup_{i \in I} A_{i} = \bigcup_{i \in 1}^{\N} A_{i} = \Z

Union Example Where $I$ Is an Interval

Another form of index sets can be that they emerge over a interval and not discretely defined values – which sounds a bit terrifying at first! 🤯 So, to delve into this, let's consider:

Let the following hold true:

I = [-1, 1]

A_{i} = \{i\} \times [0, 1]

, where $A_{i}$ is a subset of $\R^{2}$ (i.e. $A_{i} \subseteq R^{2}$ ).

If we ask what is the union of $A_{i}$ , given that $I$ is uncountably infinite, the answer would be:

\bigcup_{i \in I} A_{i} = [-1, 1] \times [0, 1]

Indexed Sets (Or How to Index Sets)

Bibliography

Table of Contents