Hi everyone! This online lesson is provided as a resource – we will also go over this material in class.
Lesson 4: Indexed Sets
Topic. This lesson covers:
- Sec 1.8: Indexed Sets
- Take unions and intersections of collections of sets indexed by the natural numbers or other sets.
Homework. There is 1 written assignment on today’s material:
- Homework: Section 1.8 p.29: 3, 5, 6, 8
- indexed sets
- index set I
When we need to talk about a lot of different sets, instead of giving each of them different names it’s convenient to keep track of them by using subscript numbers, like . 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 . We call the number and the indices.
- Unions and intersections of many sets. Suppose are sets. Then
- Notation. Given sets we define
- Definition. The index set is the set of all indices (of a collection of sets).
- Notation. If is an index set, and for each we have a corresponding set then
Examples: Indexed Sets
Example 1: Suppose and .
VIDEO: Introduction to Indexed Sets, Example 1
Example 2: Consider the following infinite list of sets:
VIDEO: Indexed Sets – Example 2
Example 3: Let the index set be the interval that is . For each number , let the set .
VIDEO: Indexed Sets – Example 3
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.
VIDEO: Introduction to Logic
Let the index set be the closed interval , that is . For each number , let the set .
(shown in red in the image below). Why do the x-values range from 1 to 2 but y-values range from 0 to 2?
. Why is this?