Name: Period: _____ Date: _______________

2.1 Basic Set Theory

Cardinal Number of set A, represents by ( ) is

Find the cardinal number of each set:

{ }

{ }

{ }

Set A is Equivalent to set B if

This Table shows the celebrities who hosted NBC’s Saturday Night Live most frequently and the number of times each starred on the show.

A = the set of the five most frequent hosts.

B = the set of the number of times each host starred on the show.

Are the sets equivalent?

Most Frequent Host of Saturday Night Live

Celebrity Number of Shows Hosted

Steve Martin 14

Alec Baldwin 12

John Goodman 12

Buck Henry 10

Chevy Chase 9

Finite set: Set A is a finite set if

Infinite set: A set whose cardinality is

Equal sets: Set A is equal to set B if

We symbolize the equality of sets A and B using the statement

If two sets are equal, then they must be equivalent!

Determine whether each statement is true or false:

{4, 8, 9} = {8, 9, 4}

{1, 3, 5} = {0, 1, 3, 5}

Find the cardinal number of each of the following sets:

{ }

{ }

{ }

Consider the following sets:

{ }

{ }

Are these sets equal? Explain:

Are these sets equivalent? Explain:

Objectives Day 2

1. Determine a set’s cardinal number.2. Recognize equivalent sets.3. Distinguish between finite and infinite sets.4. Recognize equal sets.

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Example 8Cardinality of Sets

• The cardinal number of set A, represented by n(A), is the number of distinct elements in set A. – The symbol n(A) is read “n of A.”– Repeating elements in a set neither adds new

elements to the set nor changes its cardinality.– Distinct means different, don’t put in repeats

Find the cardinal number of each set:a. A = { 7, 9, 11, 13 }

n(A) = 4b. B = { 0 }

n(B) = 1c. C = { 13, 14, 15,…,22, 23}

n(C)=11

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Equivalent Sets• Set A is equivalent to set B if set A and set B contain the

same number of elements. For equivalent sets, n(A) = n(B).

These are equivalent sets:The line with arrowheads, , indicate that each element of set A can be paired with exactly one element of set B and each element of set B can be paired with exactly one element of set A.

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One-To-One Correspondences and Equivalent Sets

• If set A and set B can be placed in a one-to-one correspondence, then A is equivalent to B: n(A) = n(B).

• If set A and set B cannot be placed in a one-to-one correspondence, then A is not equivalent to B: n(A) ≠n(B).

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Example 9Determining if Sets are Equivalent

• This Table shows the celebrities who hosted NBC’s Saturday Night Livemost frequently and the number of times each starred on the show.

A = the set of the five most frequent hosts.

B = the set of the number of times each host starred on the show.

• Are the sets equivalent?

Most Frequent Host of Saturday Night Live

Celebrity Number of Shows Hosted

Steve Martin 14Alec Baldwin 12John Goodman 12Buck Henry 10Chevy Chase 9

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Example 9 continued• Method 1: Trying to set up a One-to-One Correspondence.• Solution:The lines with the arrowheads indicate that the correspondence between the sets in not one-to-one. The elements Baldwin and Goodman from set A are both paired with the element 12 from set B. These sets are not equivalent.

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Example 9 continued

• Method 2: Counting Elements• Solution:Set A contains five distinct elements: n(A) = 5. Set Bcontains four distinct elements: n(B) = 4. Because thesets do not contain the same number of elements, theyare not equivalent.

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Finite and Infinite Sets,Equal Sets

• Finite set: Set A is a finite set if n(A) = 0 ( that is, A is the empty set) or n(A) is a natural number.

• Infinite set: A set whose cardinality is not 0 or a natural number. The set of natural numbers is assigned the infinite cardinal number 0א read “aleph-null”.

• Equal sets: Set A is equal to set B if set A and set B contain exactly the same elements, regardless of order or possible repetition of elements. We symbolize the equality of sets A and B using the statement A = B.

If two sets are equal, then they must be equivalent!

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Example 10Determining Whether Sets are Equal

Determine whether each statement is true or false:

a. { 4, 8, 9 } = { 8, 9, 4 } True

b. { 1, 3, 5 } = {0, 1, 3, 5 } False

Practice

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Practice

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Homework

• 2.1 Basic Set Concepts

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1- Basic Set Concepts Day 2 Notes.pdf2.1 - Basic Set Theory Day 2