Sets
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A Set is a collection of objects. Each object in the set is called an element or a member. Curly braces , { }, are used to list the elements in a set. The ∈ symbol is used to denote that an element is in a given set. For example: consider the set A = {2, 5, 6, 7, 9, 11}. 5 ∈ A and 8 ∈ A /

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Sets
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Sets may also be written using set builder notation of the form {x | x has property P} For example: the set of even integers can be written E = {x | x = 2k for k any integer}

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Sets
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The empty set, denoted φ, is the set that contains no elements. Do not but braces around the empty set. {φ} actually represents the set that contains the empty set.

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A Universal set, denoted U, is the set that includes all the objects being discussed.

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Sets
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The number of elements in a set B is denoted n(B). For example: n(A) = 6 and n(φ) = 0.

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Two sets are equal if they contain exactly the same elements. For example: {1, 2, 3} = {2, 3, 1}, but {1, 2, 3} = {1, 2, 4}

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Subsets
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Set A is a subset of set B, denoted A ⊆ B, if every element of A is also an element of B. A Set A is a proper subset of set B, denoted A ⊂ B, if A ⊆ B and A = B. With all of this notation it is often nice to have a diagram representing the sets.

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The diagram for sets is called a Venn diagram.

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Subsets
Here is a Venn diagram that shows that A is a subset of B.

The rectangular box represents the universal set U. The interior of each circle represents all of the elements contained in a set.

Properties of Subsets
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For any set A, φ ⊆ A and A ⊆ A. A set of n distinct elements has 2n subsets.

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Set Operations
Complement Of a Set
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Let A be any set, with U representing the universal set. Then the complement of A, denoted ∼ A, is ∼ A = {x | x ∈ A and x ∈ U }. /

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Set Operations
Intersection of Two Sets
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The intersection of sets A and B, is A ∩ B = {x | x ∈ A and x ∈ B}.

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Set Operations
Disjoint Sets
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A and B are said to be disjoint when A ∩ B = φ.

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Set Operations
Union Of Two Sets
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The union of sets A and B, is A ∪ B = {x | x ∈ A or x ∈ B}.

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Set Operations
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A∪ ∼ A = U

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∼ (A ∩ B) = (∼ A) ∪ (∼ B)

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∼ (A ∪ B) = (∼ A) ∩ (∼ B)

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Union Rule For Sets
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n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Proof:
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Let n(A) = x + y, n(B) = y + z, and n(A ∩ B) = y.

Then n(A ∪ B) = x + y + z and n(A) + n(B) − n(A ∩ B) = (x + y) + (y + z) − y = x + y + z, ⇒ n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

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Example 1
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A researcher collect data on 100 households find that
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21 have a DVD player; 56 have a videocassette recorder (VCR); 12 have both.

The researcher want to answer the following questions
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How many do not have a VCR? How many have neither a DVD players nor a VCR? How many have a DVD players but not a VCR?

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Example 1: Solution

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35 + 9 = 44 do not have a VCR. 35 have neither a DVD players nor a VCR. 9 have a DVD players but not a VCR.

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Example 2
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A group of 10 students meet to plan a school function, all are majoring in accounting or economics or both. Five of the students are economics majors and seven are majors in accounting. How many major in both subjects. Let A be a set of accounting majors. Let B be a set of economics majors.
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Use the union rule with n(A) = 5, n(B) = 7, n(A ∩ B) = 10. Find n(A ∩ B). n(A ∪ B) = n(A) + n(B) − n(A ∩ B) 10 = 5 + 7 − n(A ∩ B) n(A ∩ B) = 5 + 7 − 10 = 2

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When A and B are disjoint, then n(A ∩ B) = 0, and n(A ∪ B) = n(A) + n(B).

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Example 3
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A group of 100 workers were asked if they were college graduates and if they belonged to a union. 65 were not college graduates, 25 were nonunion college graduates and 25 were union memebers. How many of the workers were college graduates and union members? Solution: Let M = Union Member and C = College graduate n(U ) = 100 M C 15 10 25 50

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Example 4
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Given that n(A ∪ B) = 34, n(A ∩ B) = 5, n(A) = 26 and n(A ∪ B ) = 30, determine the number of element(s) in each of the disjoint regions labeled with Roman numberals in the Venn diagram below U

A x y

B z

w

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Example 5
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Use the given information to fill in the numbers of elements for each region of the Venn diagram below

A m y x w p

B n z

r C n(A ∩ B ∩ C) = 4, n(C ∩ B ) = 17, n(A ∩ B ∩ C ) = 10, n(A ∩ B) = 12, n(A ∩ C) = 8, n(B ∩ C) = 7, n(A) = 40, and n(B ∩ A ) = 20.

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Example 5: Solution
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n(A ∩ B ∩ C) = 4, n(C ∩ B ) = 17, n(A ∩ B ∩ C ) = 10, n(A ∩ B) = 12, n(A ∩ C) = 8, n(B ∩ C) = 7, n(A) = 40, and n(B ∩ A ) = 20.

A 24 4 8 4 13

B 17 3

10 C

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Example 6
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A survey of 77 freshman business students at a large university produced the following results
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25 of students read Business Week; 19 read the Wall Street Journal; 27 do not read Fortune; 11 read Business Week but not Wall Street Journal; 11 read Wall Street Journal and Fortune; 13 read Business Week and Fortune; 9 read the three.

Use the above information to answer the following questions
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How many students read none of the publications? How many read only Fortune?

How many read Business Week and the Wall Street Journal . – p.20/?? but not Fortune?

Example 6: Solution

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There are 12 students who read none of the three publications. There are 35 students who read only Fortune. There are 5 students whoread Business Week and the Wall Street Journal but not Fortune.

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Example 7
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85 students were asked if they watched the Showtime series Dexter or the HBO series Entourage. The information collected is listed below. 22 students like Dexter but not Entourage. 19 students like both Dexter and Entourage. 9 females like Dexter. 15 males like both Dexter and Entourage. 29 like neither Dexter nor Entourage. 10 females like Entourage but not Dexter. 25 females do not like Entourage. Draw a Venn diagram for this problem.

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Example 7: Solution
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22 students like Dexter but not Entourage. 19 students like both Dexter and Entourage. 9 females like Dexter. 15 males like both Dexter and Entourage. 29 like neither Dexter nor Entourage. 10 females like Entourage but not Dexter. 25 females do not like Entourage.

Dexter Entourage 17 5 15 4 5 10 9

20 Females

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