14X S16 1.1 Sets & Subsets PDF

Preview

Summary

Download 14X S16 1.1 Sets & Subsets PDF

Description

Number:______ Section 1.1 Sets & Subsets

MAT141/142 Name Austin Jonguitud (A.J.) /10

Section Objectives 1. Define sets, subsets, and the empty set. 2. Represent sets in different notations. 3. Define terminology and notation related to sets, subsets, and the empty set. 4. Recognize equivalent and equal sets. 5. Determine the number of subsets of a set. We are going to begin this class by discussing the concept of sets. Definition of a Set A set is a collection of objects, where the contents of the set can be clearly determined. The objects in the set are called elements of the set. It is important to take a moment to talk about the idea that contents of a set must be well defined, or the contents of a set can be clearly determined. Example 1 Are the following sets well defined? a) The set of teams who made the NCAA Basketball Tournament this year. This is well defined. We can go lookup which teams made the NCAA Basketball team this past year, and any will either be clearly in the set or clearly not in that set. b) The set of good basketball teams. This set is not well defined, as the adjective “good” is subjective to who is answering this question. In other words, it is not clear whether given team makes it into this set or not. We generally use capital letters to denote sets, and there are three ways we often represent sets: word description, roster method, and set-builder notation. For example, let’s look at the set of all states in the U.S. whose name starts with a C. We would usually use a letter to denote this set, so let’s use S. So, the word description of the set S would be: with C .

S=¿

the set

of states in the U.S. whose name starts

The roster method would be to use a set of braces and list the elements separated by a comma: S= {California ,Colorado ,Connecticut } The set-builder notation of a set looks like this: S= {x |x is a state∈the U . S . whose name starts with a C .} This is read as: “S is the set of all elements x such that x is a state in the U.S. whose name starts with a C.) Authors: Andrew & Kristina Burch Page 1 of 8

MAT141/142

Section 1.1 Sets & Subsets

You can use any letter you like at the beginning, so it doesn’t have to be x, but we could have used y, z, etc. ¿ ¿ ¿ or ¿ symbols for sets when using the set-builder notation or the roster method as ¿ ¿ they would have different meanings than just listing the elements. We do not use the

Example 2 Express the following set in the other 2 notations. M =¿ the set of months of the year with exactly 30 days. Roster Method: M ={ April , June , September , November } Set-Builder Notation: M ={ m|m is month with exactly 30 days } It can at times get tedious to write out each element of a set in the roster method. If the elements of the set form a particular pattern, we can use three dots (…) called an ellipsis. For example, if we were to use the roster method on the set E of all the even numbers greater than 0 and less than or equal to 20, it would look like this: E= {2, 4,6, 8, 10,12, 14,16, 18,20 } Using the ellipsis, we could simply write this as: E= {2, 4,6, … , 20 } The ellipsis indicates that the elements of the set continue in the pattern established up until 20. In mathematics, we often use symbols to avoid writing the same things over and over again. For example, given the set A = {1, 2, 3, …, 8), we have been talking about elements of this set by writing “5 is an element of A” Instead of using the words “is an element of”, we use the symbol ∈ . So, we could simply write

5 ∈ A which would be read as “5 is an element of A”.

Similarly, we can use the symbol So,

12∉ A

∉ to notate that something is not an element of a set.

would be read as “12 is not an element of A”.

Authors: Andrew & Kristina Burch

Page 2 of 8

MAT141/142

Section 1.1 Sets & Subsets One of the sets that we use and discuss often is the set of Natural Numbers. The Set of Natural Numbers N= {1, 2, 3, 4,5, 6, … } We often use inequality symbols to represent sets of natural numbers. Here is a quick review of the symbols and how they would be used in set-builder notation. Inequality Symbol

Set-Builder Notation

Roster Method

x 5 }

{ 6,7, 8, 9, …}

p≥a pis greater than∨equal ( ¿a )

{ p∨ p ∈ N ∧ p≥ 5 }

{ 5,6, 7, 8, 9,…}

a...