Chapter 2 Set Theory (Page 42 - )

Home , Associative property, Equality (mathematics), Null set, Universal set, Venn diagram

Chapter 2 Set Theory (page 42 - )

Objectives: • Specify sets using both the listing and set builder notation • Understand when sets are well defined • Use the element symbol property • Find the cardinal number of sets

Definition (sets) In mathematical terms a collection of (well defined) objects is called a set and the individual objects in this collection are called the elements or members of the set. Examples: a) S is the collection of all students in Math 1001 CRN 6977 class b) T is the set of all students in Math 1001 CRN 6977 class who are 8 feet tall d) E is the set of even natural numbers less than 2 e) B is the set of beautiful birds (Not a well-defined set) f) U is the set of all tall people (Not a well-defined set) Note: The sets in b) and d) have no elements in them. Definition: (Empty Set): A set containing no element is called an empty set or a null set. Notations: { } denotes empty set.

Representations풐풓 ∅ of Sets In general, we represent (describe) a set by listing it elements or by describing the property of the elements of the set, within curly braces. Two Methods: Listing and Set Builder

Examples of Listing Method: List the elements of the set a) N5 is the set of natural numbers less than 5 = { , , , }

b) ퟓ the set of positive integers less than 100 퐍 =ퟏ{ ퟐ, ퟑ, ퟒ, . . . , } ퟏퟎퟎ c) 퐍 = { , +, , , , } 퐍ퟏퟎퟎ ퟏ ퟐ ퟑ ퟏퟎퟎ Set Builder Method: has the general format { : ( )}, here ( ) is the property that the element x 푺 풎 ퟏ ∅ ퟓ ∆ should satisfy to be in the collection Examples, Set-builder Method: 풙 푷 풙 푷 풙 d) = { : 1001 6977 }. Here ( ) = is a student in Math 1001 CRN6977 class 푺 풙 풙 푖푠 푎 푠푡푢푑푒푛푡 푖푛 푀푎푡ℎ 퐶푅푁 푐푙푎푠푠 e) = { : 1 2} = { : 1 < < 2} Here 푷(풙) = 풙 is a real number strictly between -1 and 2 푅 푥 푥 푖푠 푎 푟푒푎푙 푛푢푚푏푒푟 푠푡푟푖푐푡푙푦 푏푒푡푤푒푒푛 − 푎푛푑 푥 − 푥 f) = { : 1 = 0} = { : = 1, 1, , } Here 푷(풙) = 풙 satisfies = 4 푇 푥 푥 푖푠 푎 푠표푙푢푡푖표푛 표푓 푡ℎ푒 푒푞푢푎푡푖표푛 푥 − 푥 푥 − −푖 푖 ퟒ Example: YouTube Videos: 푷 풙 풙 퐱 − ퟏ ퟎ . Sets and set notation: https://www.youtube.com/watch?v=01OoCH-2UWc . Set builder notation: https://www.youtube.com/watch?v=xnfUZ-NTsCE

Sets of Numbers commonly used in mathematics = = { : } = { : }

푹 = ℝ = {풙:풙 풊풔 풂 풓풆풂풍 풏풖풎풃풆풓 풙} =풙 풉풂풔: 풂 풅풆풄풊풎풂풍 풆풙풑풂풏풔풊풐 풏 , , 풂 푸= ℚ= 풙= 풙{ 풊풔: 풂 풓풂풕풊풐풏풂풍 풏풖풎풃풆풓} = { . . . �,풙 풙, 푖푠 표푓, 푡,ℎ푒 ,푓표푟푚, . 풃. .푤} ℎ 푒푟푒 풂 풃 ∈ ℤ 풃 ≠ ퟎ� { } 푰 =풁 : ℤ 풙 풙 풊풔 풂풏 풊풏풕풆품풆풓 = { , , −, ퟐ, − . ퟏ . . ퟎ } ퟏ ퟐ { } 푾= 풙=풙 풊풔: 풂 풘풉풐풍풆 풏풖풎풃 풆풓 ퟎ ퟏ = {ퟐ , ퟑ , , . . . } Definition푵 ℕ (Universal풙 풙 풊풔 풂Set)풏풂풕풖풓풂풍 풏풖풎풃풆풓 ퟏ ퟐ ퟑ The universal set is the set of all elements under consideration in a given discussion. We denote the universal set by the capital letter U.

The Element Symbol ( ) We use the symbol to stand for the phrase is an element of, and the symbol means is not an element of ∈ Example1: i) = { , ∈, , ±, , , } , ∉

푨 , ퟎread−ퟑ as ퟏퟎ0 is an element∀ 흏 푯 of A ퟎ ∈ 푨 , read as 10 is an element of A ퟏퟎ ∈ 푨, read as is an element of A 흏 ∈ 푨, read as 흏3 is not an element of A ퟑ ∉ 푨, read as h is not an element of A { { } { } { } } ii) = , 풉, ∉, 푨 , , , . Referring to set B answer the following as True or False. a) d) {{1, 0}, 0} 푩 ∅ ퟎ ∅ ∅ ퟏ ퟎ ퟎ { } b) ∅{ ∈} 퐵 e) 0, ∈ 퐵 c) {∅0} ∈ 퐵 f) 0 ∅ ∈ 퐵

∈ 퐵 ∉ 퐵 Cardinal Number Definition: The number of elements in set A is called the cardinal number of A and is denoted by ( ). A set is finite if its cardinal number is a whole number. A set is infinite if it is not finite Example 2: Find the cardinal number of 풏 푨 a) = {0, 3, 10, ±, , , } b) = { : 1 = 0} c) 푨 = { ,−{0, }, { }, {1∀, 0}휕, 0퐻} 4 d) 푇 = {푥 }푥 푖푠 푎 푠표푙푢푡푖표푛 표푓 푡ℎ푒 푒푞푢푎푡푖표푛 푥 − e) 퐵 = {∅ } ∅ ∅ f) 퐸 = {∅, , , . . . } 퐻 Example:ℕ YouTubeퟏ ퟐ ퟑ Videos: . Elements subsets and set equality: https://www.youtube.com/watch?v=kGyOrbbllEY&spfreload=10 . Introduction to Set Concepts & Venn Diagrams: https://www.youtube.com/watch?v=Jt-S9J947C8

2.2 Comparing Sets (Page 50) Objectives: • Determine when sets are equal • Know the difference between the relations subsets and proper subsets • Use Venn diagrams to illustrate sets relationships • Distinguish between the ideas of “equal” and “equivalent” sets

Sets Equality Definition (Equal sets) Two sets A and B are equal if and only if they have exactly the same members. We write = to mean A is equal to B. If A and B are not equal we write . 푨 푩

푨 ≠ 푩 Example 1: Let = { : 4} = { , , } 푨 풙 풙 푖푠 푎 푛푎푡푢푟푎푙 푛푢푚푏푒푟 퐿푒푠푠 푡ℎ푎푛 = { : 4} 푩 ퟏ ퟐ ퟑ = { : 4} 푪 풚 풚 푖푠 푎 푝표푠푖푡푖푣푒 푖푛푡푒푔푒푟 푙푒푠푠 푡ℎ푎푛 표푟 푒푞푢푎푙 푡표 = { , , , } 푫 풙 풙 푖푠 푎 푤ℎ표푙푒 푛푢푚푏푒푟 푙푒푠푠 푡ℎ푎푛 In the list above, identify the sets that are equal 푬 ퟎ ퟏ ퟐ ퟑ Solution: = and =

푨 푩 푫 푬 Subsets Definition (subset): The set A is said to be a subset of the set B if every element of A is also an element of B. We indicate this relationship by writing . If A is not a subset of B, then we write Example 2: Let = { : 4} 푨 ⊆ 푩 푨 ⊈ 푩 = { , , } 푨 풙 풙 푖푠 푎 푛푎푡푢푟푎푙 푛푢푚푏푒푟 퐿푒푠푠 푡ℎ푎푛 = { : 4} 푩 ퟏ ퟐ ퟑ = { : 4} 푪 풚 풚 푖푠 푎 푝표푠푖푡푖푣푒 푖푛푡푒푔푒푟 푙푒푠푠 푡ℎ푎푛 표푟 푒푞푢푎푙 푡표 In the set list above: and also 푫 풙 풙 푖푠 푎 푤ℎ표푙푒 푛푢푚푏푒푟 푙푒푠푠 푡ℎ푎푛 표푟 푒푞푢푎푙 푡표 but , 푨 ⊆ 푩 푩 ⊆ 푨 Proper Subset 푨 ⊆ 푪 푪 ⊈ 푨 푩 ⊆ 푪 ⊆ 푫 Definition (proper subset): A set A is said to be a proper subset of set B if but B is not a subset of A. We write to mean A is a proper subset of B. 푨 푖푠 푎 푠푢푏푠푒푡 표푓 푩 Example 3: Let = { : 4} 푨 ⊂ 푩 = { : 3} 푨 풙 풙 푖푠 푎 푛푎푡푢푟푎푙 푛푢푚푏푒푟 퐿푒푠푠 푡ℎ푎푛 = { , , , } 푪 풚 풚 푖푠 푎 푝표푠푖푡푖푣푒 푖푛푡푒푔푒푟 푙푒푠푠 푡ℎ푎푛 표푟 푒푞푢푎푙 푡표 Here , A is a proper subset of C 푬 ퟎ ퟏ ퟐ ퟑ , C is a proper subset of E 푨 ⊂ 푪 Example: YouTube Videos: 푪 ⊂ 푬 . Subsets and proper subsets: https://www.youtube.com/watch?v=1wsF9GpGd00 . Subsets and proper subsets: https://www.youtube.com/watch?v=s8FGAclojcs

Example 4: Let = { , { , }, { }, { , }, }. Referring to set B, answer the following as True or False. 푩 ∅ ퟎ ∅ ∅ ퟏ ퟎ ퟎ a) d) {{1, 0}, 0} { } b) ∅{ ⊆} 퐵 e) 0, ⊆ 퐵 c) {∅0} ⊆ 퐵 f) 0 ∅ ⊆ 퐵 Example 5:⊆ Finding퐵 all subsets of a set 퐵 Let = { , }, = { , , } and ⊆ = { , , , }. List all subsets of the sets S and T 푺 ퟎ ퟏ 푻 풂 풃 풄 푬 ퟎ ퟏ ퟐ ퟑ Solution: = { , }; the subsets of set S are , { }, { }, { , }. There are 4 = 2 subsets of S { } { } { } { } { } { 2 } 푺 = {ퟎ,ퟏ, }; the subsets of set T are∅ ퟎ, ퟏ, ퟎ, ퟏ , , , , , , , { , , }. There are 푻 = 풂 Subsets풃 풄 of T ∅ 풂 풃 풄 풂 풃 풂 풄 풃 풄 풂 풃 풄 3 ퟖ = ퟐ{ , , , }, Set E has = subsets ퟒ Number푬 of Subsetsퟎ ퟏ ퟐ ퟑof a set: ퟐ ퟏퟔ If a set has k elements, then the number of subsets of the set is given by . 풌 Equivalent Sets ퟐ Definition: Two sets A and B are equivalent, or in one to one correspondence, iff ( ) = ( ). In other words, two sets are equivalent if and only if they have the same Cardinality. 풏 푨 풏 푩 Example 6: Equivalent sets a) = { , , } and = { , , }are equivalent, Note

b) 푻 = {풂,풃 풄, , . 푩. . } andퟏ ퟐ ퟑ = { , , , , . . . }푻 are≠ 푩equivalent { } c) ℕ = {ퟏ, ퟐ, ퟑ, . . . } and 푾= :ퟎ ퟏ ퟐ ퟑ are equivalent d) ℕ = {ퟏ, ퟐ, ퟑ, . . . } and ℤ = {풙 :풙 풊풔 풂풏 풊풏풕풆품풆풓 } are equivalent e) Equℕ alퟏ setsퟐ areퟑ equivalent ℚ 풙 풙 풊풔 풂 풓풂풕풊풐풏풂풍 풏풖풎풃풆풓 Example 7: Let = 0, , , , {0} . How many subsets of A have: a) One element (list the subsets) 퐴 � 푎 푒 휋 � b) Two elements (list the subsets) c) Four elements (list the subsets)

Set Operations (page 57)

Objectives: • Perform the set operations of union, intersection, complement and difference • Understand the order in which to perform set operations • Know how to apply DeMorgan’s Laws in set theory • Use Venn diagrams to prove or disprove set theory statements • Use the Inclusion – Exclusion Principle to calculate the cardinal number of the union of two sets

Union of Sets Definition (set union ): The union of two sets A and B, written , is the set of elements that are members of A or B (or ∪ both). Using the set-builder notation, = { : } 푨 ∪ 푩 The union of more than two sets is the set of all elements belonging to at least to one of the sets. 푨 ∪ 푩 풙 풙 ∈ 푨 풐풓 풙 ∈ 푩 Example 1: = { : 7 1} = { , , } 푨 풙 풙 푖푠 푎 푛푎푡푢푟푎푙 푛푢푚푏푒푟 퐿푒푠푠 푡ℎ푎푛 푎푛푑 푔푟푒푎푡푒푟 푡ℎ푎푛 = { : 4} 푩 ퟏ ퟐ ퟑ = { : 4} 푪 풚 풚 푖푠 푎 푝표푠푖푡푖푣푒 푖푛푡푒푔푒푟 푙푒푠푠 푡ℎ푎푛 표푟 푒푞푢푎푙 푡표 Find a) c) 푫 풙 풙 푖푠 푎 푤ℎ표푙푒 푛푢푚푏푒푟 푙푒푠푠 푡ℎ푎푛 b) d) 푨 ∪ 푩 푨 ∪ 푩 ∪ 푫

푨 ∪ 푪 푪 ∪ 푩 ∪ 푫 Intersection of Sets Definition (set intersection ) The intersection of two sets A and B, written , is the set of elements common to both A and B. ∩ Using the set-builder notation, = { : } 푨 ∩ 푩 The intersection of more than two sets is the set of all elements that belongs to each of the sets. If the 푨 ∩ 푩 풙 풙 ∈ 푨 풂풏풅 풙 ∈ 푩 intersection, = , then we say A and B are disjoint.

Example 2: 푨 ∩ 푩 ∅ = { : 7 2} = { , , } 푨 풙 풙 푖푠 푎 푛푎푡푢푟푎푙 푛푢푚푏푒푟 퐿푒푠푠 푡ℎ푎푛 푎푛푑 푔푟푒푎푡푒푟 푡ℎ푎푛 = { : 4} 푩 ퟏ ퟐ ퟑ = { : 3} 푪 풚 풚 푖푠 푎 푝표푠푖푡푖푣푒 푖푛푡푒푔푒푟 푙푒푠푠 푡ℎ푎푛 표푟 푒푞푢푎푙 푡표 Find a) c) 푫 풙 풙 푖푠 푎 푤ℎ표푙푒 푛푢푚푏푒푟 푙푒푠푠 푡ℎ푎푛 b) d) 푨 ∩ 푩 푨 ∩ 푩 ∩ 푪 Example 3: Let = { , { , }, { }, { , }, } and = { , }, { }, { }, { , }, . Find: 푨 ∪ 푫 푪 ∩ 푩 ∩ 푫 a) ( ) c) 푩 ∅ ퟎ ∅ ∅ ퟏ ퟎ ퟎ 푨 � ∅ ퟎ ퟎ ∅ ퟎ ퟏ ퟏ� b) ( ) d) ( ) 풏 푩 푨 ∩ 푩

풏 푨 푛 퐴 ∩ 퐵 Example: YouTube Videos: . Intersection and union of sets: https://www.youtube.com/watch?v=jAfNg3ylZAI

Set Complement Definition ( , ) If A is a subset′ of the universal set U, the complement of A is the set of elements of U that are not 푨 푨 풑풓풊풎풆 표푟 푨 풄풐풎풑풍풆풎풆풏풕 elements of A. This set is denoted by . Using the set-builder notation, = { : } Example 4: Using Venn diagram: ′ 푨′ 푨 풙 풙 ∈ 푼 풃풖풕 풙 ∉ 푨 a) Show that, if , then = b) Show that, if , then =

푨 ⊆ 푩 푨 ∩ 푩 푨 푨 ⊆ 푩 푨 ∪ 푩 푩 Example 5: Let = {0, 1, 2, 3, . . . , 10} and = { : 7 2} 푼 = { , , } = { : 4} 푨 풙 풙 푖푠 푎 푛푎푡푢푟푎푙 푛푢푚푏푒푟 퐿푒푠푠 푡ℎ푎푛 푎푛푑 푔푟푒푎푡푒푟 푡ℎ푎푛 = { : 3} 푩 ퟏ ퟐ ퟑ 푪 풚 풚 푖푠 푎 푝표푠푖푡푖푣푒 푖푛푡푒푔푒푟 푙푒푠푠 푡ℎ푎푛 표푟 푒푞푢푎푙 푡표 Find a) b) c) d) 푫 ′풙 풙 푖푠 푎 푤ℎ표푙푒 푛푢푚푏푒푟′ 푙푒푠푠 푡ℎ푎푛 ′ ′ e) 푨 f) 푩 g) 푪( ) h) 푫( ) ′ ′ Set Difference푨 ∪ 푩′ 푪 ∪ 푩′ 푪 ∪ 푫 ′ 푩 ∩ 푪 ′ Definition (B – A, B less A) The difference of sets B and A is the set of elements that are in B but not in A. This set is denoted by . Using the set-builder notation, = { : }

Example푩 − 푨6: Let = { : 푩 − 푨 풙 풙 ∈ 7푩 풂풏풅 풙 ∉ 푨 2} = { , , } = { : 4} 푨 풙 풙 푖푠 푎 푛푎푡푢푟푎푙 푛푢푚푏푒푟 퐿푒푠푠 푡ℎ푎푛 푎푛푑 푔푟푒푎푡푒푟 푡ℎ푎푛 = { : 3} 푩 ퟏ ퟐ ퟑ 푪 풚 풚 푖푠 푎 푝표푠푖푡푖푣푒 푖푛푡푒푔푒푟 푙푒푠푠 푡ℎ푎푛 표푟 푒푞푢푎푙 푡표 Find: a) b) c) d) ( ) 푫 풙 풙 푖푠 푎 푤ℎ표푙푒 푛푢푚푏푒푟 푙푒푠푠 푡ℎ푎푛

푨 − 푩 푫 − 푪 푪 − 푩 푨 − 푩 − 푪 Venn Diagrams The Universal Set ( ) is represented by a rectangle. The shaded regions represent, respectively, the union, intersection, difference and complement of the sets and . 퐔 푨 푩 a) b) B A B A

푨 ∪ 푩 푨 ∩ 푩

Example: YouTube Videos: . Intersection and union of sets: https://www.youtube.com/watch?v=jAfNg3ylZAI

c) B d) B A A ′ 푨 − 푩 푨

Example: YouTube Videos: . Difference or relative complement: https://www.youtube.com/watch?v=2B4EBvVvf9w . Absolute complement: https://www.youtube.com/watch?v=GVZUpOm3XUg

Order of Set Operations Just as we perform arithmetic operations in a certain order, set notations specifies the order in which we perform set operations.

1. Just like with numbers, we always do anything in parentheses first. If there is more than one set of parentheses, we work from the inside out.

2. Union, intersection, and difference operations are all equal in the order of precedence. So, if we have more than one of these at a time, we have to use parentheses to indicate which of these operations should be done first.

Properties of Set Operations: If A, B and C be sets, then a) = Commutative property of union b) = Commutative property of intersection 푨 ∪ 푩 푩 ∪ 푨 c) ( ) = ( ) Associative property of union 푨 ∩ 푩 푩 ∩ 푨 d) ( ) = ( ) Associative property of intersection 푨 ∪ 푩 ∪ 푪 푨 ∪ 푩 ∪ 푪 e) ( ) = ( ) ( ) Distributive property of union over intersection 푨 ∩ 푩 ∩ 푪 푨 ∩ 푩 ∩ 푪 f) ( ) = ( ) ( ) Distributive property of intersection over union 푨 ∩ 푩 ∪ 푪 푨 ∪ 푪 ∩ 푩 ∪ 푪 Proof: Use Venn diagram 푨 ∪ 푩 ∩ 푪 푨 ∩ 푪 ∪ 푩 ∩ 푪

Note that: , , , , and so on, without parenthesis indicating which to do first, are ambiguous. 푨 − 푩 − 푪 푨 ∪ 푩 ∩ 푪 푨 − 푩 ∪ 푪 푨 ∩ 푩 ∪ 푪 푨 ∩ 푩 − 푪 Example 7: Let = {0, 1, 2, 3, . . . , 10} be the universal set and = { : 9 2} = { , , , , , } 푼 = { , , } = { : 4} = { , , , } 푨 풙 풙 푖푠 푎 푛푎푡푢푟푎푙 푛푢푚푏푒푟 퐿푒푠푠 푡ℎ푎푛 푎푛푑 푔푟푒푎푡푒푟 푡ℎ푎푛 ퟑ ퟒ ퟓ ퟔ ퟕ ퟖ = { : 3} 푩 ퟏ ퟐ ퟑ 푪 풚 풚 푖푠 푎 푝표푠푖푡푖푣푒 푖푛푡푒푔푒푟 푙푒푠푠 푡ℎ푎푛 표푟 푒푞푢푎푙 푡표 ퟏ ퟐ ퟑ ퟒ Find푫 a) 풙( 풙 푖푠 푎) 푤ℎ표푙푒 푛푢푚푏푒푟b) 푙푒푠푠( 푡ℎ푎푛) c) ( ) j) ( ) d) ( ) e) ( ) f) ( ) 퐴 − 퐵 − 퐶 퐴 − 퐵 − 퐶 퐴 ∩ 퐵 ∪ 퐶 퐴 ∪ 퐵 ′ g) h) ( ) i) 퐴 ∩ 퐵 ∪ 퐶 퐴 ∩ 퐵 − 퐶 퐴 ∩ 퐵 − 퐶

퐴′ ∩ 퐵′ 퐴 ∩ 퐵 ′ 퐴′ ∪ 퐵′ 7

DeMorgan’s Laws: If A and B are sets, then a) ( ) = and b) ( )′ = 푨 ∪ 푩 푨′ ∩ 푩′ ′ 푨 ∩ 푩 푨′ ∪ 푩′ The inclusion-Exclusion Principle If A and B are sets, then ( ) = ( ) + ( ) ( )

Example 8: Let = { 0, 1,풏2,푨3∪, 4푩, 5, 6풏, 푨, , 풏, 푩, −, 풏, 푨 ∩}, the푩 universal set, = {0, 1, 2, 3, 4}, { } { } { } = { , , 1, 2퐔, 3, , }, = 0, , , 푎 푏, 푐=푑 0,푒 1,푓 3,푔 , 5, 6 , and = 퐴0, 1, 6, , . Find:퐵 a)푎 푏 푐 푒 퐶 b) ( 푏 푔) 푑 ( 퐷 ) 푒 c) ( 퐸 ) ( )푎 푓 ′ ′ d) 푨Verify∩ �푨 that∩ 푩 �( ) = 푨( ∪) 푩+ ∩( 푫) − 푪( ) 푬 ∩ 푪 ′ ∩ 푫 ∪ 푩

f) Verify DeMorgan’s풏 푨 ∪ 푩 Laws풏 for푨 the풏 sets푩 B− and풏 푨 D∩ 푩

Example: YouTube Videos: . Bringing the set operations together: https://www.youtube.com/watch?v=OCNXS_m1HWU

Solving Survey Problems with Venn Diagrams Objectives: • Label sets in Venn diagrams with various names • Use Venn diagrams to solve survey problems • Understand how to handle contradictory information in survey problems

Example: YouTube Videos . Solving problems using Venn Diagrams: https://www.youtube.com/watch?v=MassxXy8iko

Example 1: Describe the following sets using Venn diagram a) Let = { , , , , , , , , , , , , , } be the universal set = { , , , , , , } and = { , , , , , , }. 퐔 ퟎ ퟏ ퟐ ퟑ ퟒ ퟓ ퟔ 풂 풃 풄 풅 풆 풇 품 푨 ퟎ ퟏ ퟐ ퟑ ퟒ 풂 풄 푩 풂 풃 ퟏ ퟐ ퟑ 풄 풆 Solutions: U A B

a, 1, 2, 0, 4 3, c b, e

5, 6, d, f, g

b) Let = { , , , , , , , , , , , , , }, the universal set = { , , , , , , }, = { , , , , , , }, and = { , , , , , } 퐔 ퟎ ퟏ ퟐ ퟑ ퟒ ퟓ ퟔ 풂 풃 풄 풅 풆 풇 품

푪 ퟎ ퟏ ퟐ 풃 품 풅 풄 푫 ퟎ ퟏ ퟑ 풆 풃 ퟓ ퟔ 푬 ퟎ ퟏ ퟔ 풂 풃 풇

U D C

3, 5, e 2, g, d, c

0, 1, b

4 a E

Example 2: Represent Using Venn diagram. Let = { , , , , . . . , } be the universal set and { } { } = : 8퐔 ퟎ ퟏ ퟐ ퟑ 1 ퟏퟎ= , , , , , { } { } 푨 = 풙: 풙 푖푠 푎 푛푎푡푢푟푎푙 푛푢푚푏푒푟 퐿푒푠푠 푡ℎ푎푛 푎푛푑 푔푟푒푎푡푒푟 푡 ℎ푎푛 = , ퟐ, ퟑ, ,ퟒ , ퟓ ퟔ ퟕ = { : 4} = { , , , } 푩 풙 풙 푖푠 푎푛 푒푣푒푛 푤ℎ표푙푒 푛푢푚푏푒푟 푙푒푠푠 푡ℎ푎푛 표푟 푒푞푢푎푙 푡표 ퟏퟎ ퟎ ퟐ ퟒ ퟔ ퟖ ퟏퟎ

Solution:푪 풚 풚 푖푠 푎 푝표푠푖푡푖푣푒 푖푛푡푒푔푒푟 푙푒푠푠 푡ℎ푎푛 표푟 푒푞푢푎푙 푡표 ퟏ ퟐ ퟑ ퟒ

= { , , , , . . . , } , = { , , , , , }, = { , , , , , }, = { , , , }

퐔 ퟎ ퟏ ퟐ ퟑ ퟏퟎ 푨 ퟐ ퟑ ퟒ ퟓ ퟔ ퟕ 푩 ퟎ ퟐ ퟒ ퟔ ퟖ ퟏퟎ 푪 ퟏ ퟐ ퟑ ퟒ

U A B 6

5, 7 0, 8, 10

2, 4 3

9 1 C

Example 3: For each of the following, using the given information, find the number of elements in set A, B, and C. a) = { }, ( ) = 5, ( ) = 3, ( ) = 7, ( ) = 2, ( ) = 14

b) 퐴(∩ 퐵 ) = 5푛, 퐴(∩ 퐶 푛) =퐵2∩, 퐶 ( 푛) =퐶 −6, 퐴 ( 푛)퐴=−10퐶, ( 푛 푈) = 23, 푛(퐴 ∩ 퐵) = 7, 푛(퐴 ∩ 퐵 ∩ 퐶) = 31 푛 퐶 ∩ 퐵 푛 퐵 − 퐴 푛 퐵 ∪ 퐶 Solution푛: 퐴Use∩ 퐶Venn diagram푛 퐴 ∩ 퐵 ∪ 퐶

Survey Problems Example 4: The National Resource Defense Council, which was instrumental in drafting California’s Global Warming Solution Act, believes that by using technology properly, we can cut U.S. global warming solution by half. Three of the solutions proposed by NRDC are using energy-efficient appliances, driving energy-efficient cars, and using renewable energy sources. Assume that you surveyed 100 members of Congress to determine which solutions they favored funding and obtained the following results: a) 12 favored the increased use of renewable energy source only b) 20 recommended funding both energy-efficient appliances and renewable energy sources c) 22 favored funding both energy-efficient cars and increased use of renewable energy sources d) 14 want to fund all three areas From this information, determine the total number who favored increased funding for renewable energy. Solution: Good Notations; we will represent this information using sets = = { : }

퐸퐴 = 푨 = {풙: 풙 풇풂풗풐풓풔 풆풏풆풓품풚 풆풇풇풊풄풊풆풏풕 − 풂풑풑풍풊} 풂풏풄풆풔 퐸퐶 = 푪 = {풙: 풙 풇풂풗풐풓풔 풆풏풆풓품풚 풆풇풇풊풄풊풆풏풕 − 풄풂풓풔 } The푅퐸 conditions푹 from풙 풙 풇풂풗풐풓풔a) – d) can풓풆풏풖풂풃풍풆 be written풆풏풆풓품풚 using set풔풐풖풓풄풆풔 notations as follows: ( ) = , ( ) = , ( ) = , ( ) = ′ ′ Venn풏 푹diagram∩ 푨 ∩ 푪representationsퟏퟐ 풏 푨of∩ the푹 sets:ퟐퟎ 풏 푪 ∩ 푹 ퟐퟐ 풏 푨 ∩ 푪 ∩ 푹 ퟏퟒ

U A C

6 8 12

The total number who favored renewable energy is + + + =

ퟔ ퟏퟒ ퟖ ퟏퟐ ퟒퟎ Example 5: Fitness Survey Personal Fitness Magazine surveyed a group of young adults regarding their exercise programs and the following results are obtained: a) 3 were using resistance training, Tae Bo, and Pilates to improve their fitness. b) 5were using resistance training and Tae Bo c) 12 were using Tae Bo and Pilates d) 8 were using resistance training and Pilates e) 15 were using resistance training only f) 30 were using Tae Bo g) 17 were using Pilates but not Tae Bo. h) 14 were using something other than these three types of workouts 1) How many people were surveyed? 2) How many people were using only Tae Bo? 3) How many people were using Pilates but not resistance training?

Solution: Using set Notations we will represent this information using sets The group of young adults surveyed is the universal set U containing three subsets. = { : } = { : } 푷 풙 풙 풊풔 풂 풚풐풖풏품 풂풅풖풍풕풔 풖풔풊풏품 푷풊풍풂풕풆풔 = { : } 푻 풙 풙 풊풔 풂 풚풐풖풏품 풂풅풖풍풕 풖풔풊풏품 푻풂풆 푩풐 The conditions from a) – h) can be written using set notations as follows: 푹 풙 풙 풊풔 풂 풚풐풖풏품 풂풅풖풍풕 풖풔풊풏품 푹풆풔풊풔풕풂풏풄풆 푻풓풂풊풏풊풏품 ( ) = ( ) = ( ) = ( ) = ,

풏(푹 ∩ 푻 ∩ 푷 ) =ퟑ , 풏( 푹)∩=푻 , ퟓ 풏 (푻 ∩ 푷 ) =ퟏퟐ , 풏(푹 ∩ 푷 ퟖ ) = , 풏 푹 ∩ 푷′ ∩ 푻′ ퟏퟓ 풏 푻 ퟑퟔ 풏 푷 ∩ 푻′ ퟏퟕ 풏 푷′ ∩ 푻′ ∩ 푹′ ퟏퟒ Venn diagram representation of the sets:

U P 12 9 T

16 3

5 2

14 15

1) How many people were surveyed? + + + + = 2) How many people were using only Tae Bo? 16 ퟏퟒ ퟏퟐ ퟏퟓ ퟏퟔ ퟏퟗ ퟕퟔ 3) How many people were using Pilates but not resistance training? + =

ퟗ ퟏퟐ ퟐퟏ 11

Example 6: Social Media Survey In a survey of 100 college students, the following information is obtained regarding their use of several social media sites: a) 21 use Facebook, LinkedIn, and Twitter b) 32 use both LinkedIn and Twitter c) 44 use Facebook and Twitter d) 31 use Facebook but not LinkedIn e) 78 use Facebook or Twitter f) 12 use only Twitter g) 5 use none of these three sites How many of those surveyed use Facebook? How many do not use Twitter?

Solution: Using set Notations; we will represent this information using sets The 100 group of students surveyed is the universal set U containing three subsets. = { : } = { : } 푭 풙 풙 풊풔 풂 풔풕풖풅풆풏풕 풖풔풊풏품 푭풂풄풆풃풐풐풌 = { : } 푳 풙 풙 풊풔 풂 풔풕풖풅풆풏풕풔 풖풔풊풏품 푳풊풏풌풆풅푰풏

The푻 condi풙 tions풙 풊풔 풂from풔풕풖풅풆풏풕풔 a) – g) can풖풔풊풏품 be written푻풘풊풕풕풆풓 using set notations as follows: ( ) = ( ) = ( ) = ( ) = ,

풏 (푭 ∩ 푳 ∩) =푻 ,ퟐퟏ 풏( 푳 ∩ 푻 ퟑퟐ) = ,풏 푭 ∩ 푻 ퟒퟒ ( 풏 푭 ∩ 푳)′ = ퟑퟏ Venn풏 푭 diagram∪ 푻 ퟕퟖrepresentation풏 of푻 the∩ 푳 sets:′ ∩ 푭 ′ ퟏퟐ 풏 푭′ ∩ 푻′ ∩ 푳′ ퟓ

U F

3 L 8

17 21

23 11

5 12

How many of those surveyed use Facebook? + + + = How many do not use Twitter? + + + = ퟖ ퟑ ퟐퟏ ퟐퟑ ퟓퟓ

ퟖ ퟑ ퟏퟕ ퟓ ퟑퟑ

Example 7: Survey of TV Preferences (Example 4 Page 70) A television survey conducted a market survey to determine the evening viewing preferences of people in the 18 – 25 age bracket. The following information was obtained: a) 3 prefer a reality show early on weekdays b) 14 want to watch TV early on weekdays c) 21 want to see reality shows early d) 8 want reality shows on weekdays e) 31 want to watch TV on weekdays f) 36 want to watch TV early g) 40 want to see reality shows h) 13 prefer late, weekends shows that are not reality shows How many people do not want to see reality shows? How many prefer to watch TV on the weekend. Solution: Using set Notations; we will represent this information using sets The set of people surveyed is a universal set containing three subsets. = { : } = { : } 푾 풙 풙 풘풂풏풕풔 풕풐 풘풂풕풄풉 푻푽 풐풏 풘풆풆풌풅풂풚풔 = { : } 푬 풙 풙 풘풂풏풕풔 풕풐 풘풂풕풄풉 풆풂풓풍풚 푻푽 풔풉풐풘풔

The푹 conditions풙 풙 풘풂풏풕풔 from 풕풐a) 풘풂풕풄풉– h) can풓풆풂풍풊풕풚 be written푻푽 using풔풉풐풘풔 set notations as follows: ( ) = , ( ) = , ( ) = , ( ) = , ( ) = ,

풏(푾′ ∩ 푬)′=∩ 푹′, (ퟏퟑ 풏 )푹= ퟒퟎ, 풏(푬 ퟑퟔ )풏=푾 ퟑퟏ 풏 푹 ∩ 푾 ퟖ Venn풏 diagram푹 ∩ 푬 representationퟐퟏ 풏 푬 ∩ 푾 of theퟏퟒ sets: 풏 푹 ∩ 푾 ∩ 푬 ퟑ

U W E 12 11 4 3

5 18

13 14

Number of people who do not want to see the reality show is + + + = Number of people who watch the TV on weekends is + + + = ퟏퟐ ퟏퟏ ퟒ ퟏퟑ ퟒퟎ

ퟒ ퟏퟖ ퟏퟒ ퟏퟑ ퟒퟗ

Example 8: online music survey Pandora.com surveyed a group of subscribers regarding which online music channels they use on a regular basis. The following information summarizes their answers: a) 7 listened to rap, heavy metal and alternative rock b) 10 listened to rap and heavy metal c) 13 listened to heavy metal and alternative rock d) 12 listened to rap and alternative rock e) 17 listened to rap f) 24 listened to heavy metal g) 22 listened to alternative rock 1) How many people were surveyed? 2) How many people listened to either rap or alternative rock? 3) How many listened to heavy metal only?

Example 9: Academic Services Survey The Dean of Academic Services surveyed a group of students about which support services they were using to help them improve their academic performance and found the following results. a) 5 were using office hours, tutoring, and online study groups to improve their grades b) 16 were using office hours and tutoring c) 28 were using tutoring d) 14 were using tutoring and online study groups e) 8 were using office hours and online study groups but not tutoring f) 23 were using office hours but not tutoring g) 18 were using only online study groups h) 37 were using none of these services 1) How many students were surveyed? 2) How many students were using only office hours? 3) How many students were using online study groups?

Example 9: News Sources Survey A survey of young adults was taken to determine which of the various sources they use to obtain news. Of the 36 people who use the internet, 13 use the internet only to learn the news. Of the 48 people who use the newspaper, 11 use the newspaper only for the news coverage. There are 23 people who use newspapers, the internet and the television for the news coverage. From this information, determine how many use both the Internet and television to learn the news.