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) the set of positive integers less than 100 퐍ퟓ 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 1 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 ퟏ ퟎ ∈ 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 2.2 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 3 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) ℕEqualퟏ 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) 4 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.