Lesson 4: Indexed Sets

Hi everyone! This online lesson is provided as a resource – we will also go over this material in class.

Vocabulary

• indices
• indexed sets
• index set I

Indexed Sets

Introduction

When we need to talk about a lot of different sets, instead of giving each of them different names $A, B, C, D, E, F$ it’s convenient to keep track of them by using subscript numbers, like $A_{1}, A_{2}, A_{3}, A_{4}, A_{5}$. The subscript numbers are called indexes (“indices”), and the sets are called indexed sets. 

Definitions and Theorems

• Indexed sets are sets that are distinguished by attaching subscript numbers (instead of using different letters), such as $A_{1}, A_{2}, A_{3}, A_{4}, A_{5}$. We call the number $1,2,3,4$ and $5$ the indices.
• Unions and intersections of many sets. Suppose $A_{1}, A_{2}, A_{3}, \ldots, A_{n}$ are sets. Then
  • $A_{1} \cup A_{2} \cup A_{3} \cup \ldots \cup A_{n}=\left\{x: x \in A_{i}\text{ for at least one set } A_{i}, \text{ for }1 \leq i \leq n\right\}$
  • $A_{1} \cap A_{2} \cap A_{3} \cap \ldots \cap A_{n}=\left\{x: x \in A_{i} \text{ for every set } A_{i}, \text{ for } 1 \leq i \leq n\right\}$
• Notation. Given sets $A_{1}, A_{2}, A_{3}, \ldots, A_{n},$ we define $$\bigcup_{i=1}^{n} A_{i}=A_{1} \cup A_{2} \cup A_{3} \cup \ldots \cup A_{n} \text{ and }$$
  $$\bigcap_{i=1}^{n} A_{i}=A_{1} \cap A_{2} \cap A_{3} \cap \ldots \cap A_{n}$$
• Definition. The index set $I$ is the set of all indices (of a collection of sets).
• Notation. If $I$ is an index set, and for each $\alpha \in I$ we have a corresponding set $A_{\alpha},$ then
  • $\bigcup_{\alpha \in I} A_{\alpha}=\left\{x: x \in A_{\alpha}\text{ for at least one set } A_{\alpha} \text{ with }\alpha \in I\right\}$
  • $\bigcap_{\alpha \in I} A_{\alpha}=\left\{x: x \in A_{\alpha}\text{ for every set } A_{\alpha} \text{ with }\alpha \in I\right\}$

Examples: Indexed Sets

Example 1: Suppose $ A_{1}=\{0,2,5\}, A_{2}=\{1,2,5\}$ and $A_{3}=\{2,5,7\}$.
Find $\bigcup_{i=1}^{3} A_{i} \text { and } \bigcap_{i=1}^{3} A_{i}$

Example 2: Consider the following infinite list of sets:
$$A_{1}={-1,0,1}, A_{2}={-2,0,2}, A_{3}={-3,0,3}, \ldots, A_{i}={-i, 0, i}, \ldots$$
Find $\bigcup A_{i} \text { and } \bigcap_{i=1}^{\infty} A_{i}$

Example 3: Let the index set $I$ be the interval $[4,5),$ that is $I=[4,5)]=\{x: 4 \leq x<5\}$. For each number $\alpha \in I$, let the set $A_{\alpha}=\{x \in \mathbb{R}: \alpha \leq x \leq 6\}$.
$$\text { Find } \bigcup_{\alpha \in[4,5)} A_{\alpha} \text { and } \bigcap_{\alpha \in[4,5)} A_{\alpha}$$

Introduction to Logic

The word logic refers to the way that humans reason – how we combine old information to deduce new information.  It is nothing exotic – you use logic all the time in your everyday life, and certainly when you do mathematics.  In Chapter 2, we will be looking at this familiar tool and studying its rules.