the cedar ledge

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

Table of Contents

    1. Reading Motivation
    2. Building an Intuition for Index Sets
      1. Union Example Where II Is a Finite Set of Integers
      2. Union Example Where II Is a Countably Infinite Set of Integers
      3. Union Example Where II Is an Interval
  1. How To Cite
  2. References:
  3. 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 II can be nearly anything. The following examples build intuition to that statement:

Union Example Where II 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 II:

Let the following hold:

I={1,2,3} I = \{1, 2, 3\} A1={0,1,2,3,4,5} A_{1} = \{0, 1, 2, 3, 4, 5\} A2={3,2,1,0,1,2} A_{2} = \{-3, -2, -1, 0, 1, 2\} A3={4,2,0,2,4} A_{3} = \{-4, -2, 0, 2, 4\}

The intersection of these three sets can be denoted as follows: iIAi\bigcup_{i \in I} A_{i}. This syntax is also synonymous with the following syntax that may be more easily read: i=13Ai\bigcup_{i=1}^{3} A_{i}. Either syntax means loosely, "create the union between sets A1A_{1} through AnA_{n}." The solution to this problem would be:

iIAi=i=13Ai={4,3,2,1,0,1,2,3,4,5} \bigcup_{i \in I} A_{i} = \bigcup_{i=1}^{3} A_{i} = \{-4, -3, -2, -1, 0, 1, 2,3, 4, 5\}

.

NOTE: A1,A2,A3A_{1}, A_{2}, A_{3} are independent sets and do not form a family of sets.

Union Example Where II 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 II:

Let the following hold:

I=N={1,2,3,} I = \N = \{1, 2, 3, \dots \} Ai={i,0,i} A_{i} = \{-i, 0, i\}

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

:Ai=i,0,i={A1=1,0,1,i=1 A2=2,0,2,i=2 A3=3,0,3,i=3 ––: 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:

iIAi=i1NAi=Z \bigcup_{i \in I} A_{i} = \bigcup_{i \in 1}^{\N} A_{i} = \Z

Union Example Where II 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] I = [-1, 1] Ai={i}×[0,1] A_{i} = \{i\} \times [0, 1]

, where AiA_{i} is a subset of R2\mathbb R^{2} (i.e. AiR2A_{i} \subseteq R^{2}).

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

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

How To Cite

Zelko, Jacob. Indexed Sets (Or How to Index Sets). https://jacobzelko.com/09302022040126-indexed-sets. September 30 2022.

References:

Discussion:

CC BY-SA 4.0 Jacob Zelko. Last modified: January 17, 2023. Website built with Franklin.jl and the Julia programming language.