STAT2125 Module 1. Set Theory_242501.pdf

Mathematical Statistics I CLARISSA JEWEL B. ALOTA STAT 2125: Mathematical Statistics I 1st Semester A.Y. 2024-2025 CENTRALLUZONSTATEUNIVERSITY DEPARTMENT of S T A T I S T ICS Statistics Fields of The study of Sta...

Mathematical Statistics I CLARISSA JEWEL B. ALOTA STAT 2125: Mathematical Statistics I 1st Semester A.Y. 2024-2025 CENTRALLUZONSTATEUNIVERSITY DEPARTMENT of S T A T I S T ICS Statistics Fields of The study of Statistics can be divided into two fields: Theory and Methods. Statistical Methods Statistics Statistical Theory Basic Concepts in Statistics | 2 STATISTICS DEPARTMENT of  Fields of Statistics 1. Statistical Methods - These are procedures and techniques used in the collection, presentation, analysis, and interpretation of data. Also referred as Applied Statistics. Procedure – a fixed, step-by-step sequence of activities that must be followed in the same order to correctly performs a task. Technique – is a practical method, skill, or art applied to a particular task.  Basic Concepts in Statistics | 3 DEPARTMENT of STATISTICS Fields of Statistics 2. Statistical Theory - Deals with the development and exposition of theories that serve as bases of statistical methods. Also referred as Theoretical Statistics or Mathematical Statistics. Theory is defined as a set of statements or principles devised to explain a group of facts or phenomena,  especially one that has been repeatedly tested or is widely accepted and can be used to make predictions about natural phenomena. Basic Concepts in Statistics | 4 1. Introduction to Probability Theory JOMEL R. ALANZALON & CLARISSA JEWEL B. ALOTA STAT 2125: Mathematical Statistics I 1st Semester A.Y. 2024-2025 CENTRALLUZONSTATEUNIVERSITY 1.1 Set Theory JOMEL R. ALANZALON & CLARISSA JEWEL B. ALOTA STAT 2125: Mathematical Statistics I 1st Semester A.Y. 2024-2025 CENTRALLUZONSTATEUNIVERSITY DEPARTMENT of S T A T I S T I C S Elements 1.1.1 Set, Universe, and Definitions A set is a well-defined collection of objects, and they are called the elements or members of the set. Capital letters, , , , , …, denote sets Lowercase letters, , , , , …, denote elements of sets Synonyms for set are class, collection, and family Set Theory | 7 DEPARTMENT of STATISTIC S  Examples: Set or Not Set a. The collection of colors of the rainbow S b. The first five nonnegative integers S c. = {2, 3, 5, 7} S d. The collection of good BS Statistics students at CLSU NS e. Group of young faculty members from CLSU NS Set Theory | 8 DEPARTMENT of STATISTIC S Definitions All sets under investigation in any application of set theory are assumed to be contained in some large fixed set called the universal set or universe of discourse.  Usually denoted by or. Set Theory | 9 DEPARTMENT of S TATISTICS Sets 1.1.2 Specifying In Roster notation, we give a comprehensive listing of the elements in the set.  In Set-builder Notation or Rule Method, we define a set variable, followed by verbal description of the category of elements and the rule used to determine which elements are in the set and which are not. Example: Let be the set of positive odd numbers less than 10 Roster Notation: = 1, 3, 5, 7, 9 Set-Builder Notation: = is a positive odd number less than 10} Set Theory | 10 DEPARTMENT of STATISTIC S Other Examples: 1. Let be the set of vowels of the English Alphabet Roster: = , , , , Set-Builder: = is a vowel of the English Alphabet} 2. Let be the set of positive integers less than 10 Roster: = 1, 2, 3, 4, 5, 6, 7, 8, 9 Set-Builder: = is a positive integer less than 10} or = ∈ ℤ, 1 ≤ ≤ 9} 3. = is a province in the Philippines} Set Theory | 11 DEPARTMENT of S T A T I S T I CS 1.1.3 Kinds  of Sets Sets can be finite or infinite. A set is finite if is empty or if consists of exactly elements where is a positive integer; otherwise set is infinite. The number of elements of a finite set is called the order or cardinal number of set and is denoted by or | |. Examples: 1. = −8, −1, 4, 1998 4. = is a positive 2. = is a month integer} of the year} 3. = {0, 1, 2, = = 3, 4 , 5} = infinite set  Set Theory | 12 DEPARTMENT of STATISTIC S A set with no element is called the empty set or null set, and is denoted by ∅ or. Thus, ∅ = 0. Examples: Set of people living in the sun Collection of triangle having four sides 2 = < 0, ∈ ℝ} Note: The zero set, 0 , is not a null set.  Set Theory | 13 DEPARTMENT of S T A T I S T I C S Between Sets 1.1.4 Relationship We write ∈ , if is an element in the set. The membership symbol ∈ is used to say that an object is a member of the given set. On the other hand, the symbol ∉ is used to say that an object is not in the given set. Illustration: 1. Consider = {blue, red, yellow}, we can write red ∈. On the  other hand, green ∉. 2. Let = {0, 1 , 2}, here 1 ∉ but instead 1 ∈ Set Theory | 14 DEPARTMENT of STATISTIC S Subset Set is a subset of set , written ⊂ or ⊃ , if every element of is also an element of , that is If ⊂ , then ∈ implies ∈. On the other hand, we write ⊄ to mean that there is at  least one element of that is not in. Proper Subset (⊂) and Improper Subset (⊆) Any set must be a subset of itself, that is ⊆. The empty set Ø is always a subset of any set. Set Theory | 15 DEPARTMENT of STATISTIC S Example: Subset Consider the sets: = , , , , = , , ,  = is a letter in the English alphabet} Then, ⊂ ⊂ ⊄ ⊄ ⊄ ⊄ Set Theory | 16 DEPARTMENT of STATISTIC S Equal Sets Set and set are equal, denoted by = , if they have exactly the same elements or if each set is a subset of each other, that is = if and only if ⊆ and ⊆. Equivalent Sets Any set is equivalent to set , denoted by ∼ , if they have the same number of elements or cardinality, that is ∼ implies = ( ). Set Theory | 17  DEPARTMENT of STATISTIC S Example: Equal Sets and Equivalent Sets Consider the sets: = , , , , = | is a vowel of the English Alphabet = −2, 0, 1, 7, 12 = { , , , } Then, = ∼ ∼  ≠ ≁ ≁ Set Theory | 18 DEPARTMENT of STATISTIC S Theorem 1.1 Let , , be any sets. Then: ⊆ If ⊆ and ⊆ , then =. If ⊆ and ⊆ , then ⊆.  Theorem 1.2 For any set , we have Ø ⊆ ⊆ Set Theory | 19 DEPARTMENT of S T A T I S T I C S Operations 1.1.5 Venn Diagram and Set A Venn diagram is a pictorial representation of sets where sets are represented by enclosed areas in the plane. The universal set is represented by the points in a rectangle, and the other sets are represented by disks lying within the rectangle. Ω Set Theory | 20 DEPARTMENT of STATISTICS 1. Union of Sets  The union of two sets and , denoted by ∪ , is the set of all element which belongs to or to. Example: = {1, 2, 3, 4, 5} = {2, 4, 6, 8} Then, Set Operations ∪ = {1, 2, 3, 4, 5, 6, 8} ∪ = ∈ ∈ }  Set Theory | 21 DEPARTMENT of STATISTICS Set Operations 2. Intersection of Sets The intersection of two sets and , denoted by ∩ , is the set of all element which belongs to both and. Example: = {1, 2, 3, 4, 5} = {2, 4, 6, 8} Then,  ∩ = {2, 4} ∩ = ∈ ∈ } Set Theory | 22 DEPARTMENT of STATISTIC S Two sets and are said to be mutually exclusive or disjoint if they have no elements in common. Thus, ∩ = Ø. Example: = { , , , } = { , , } ∩ = Ø  ∴ and are disjoint sets. Set Theory | 23 DEPARTMENT of STATISTIC S Properties of Union and Intersection 1. Every element in ∩ belongs to both and , hence belongs to and belongs to. Thus, ∩ is a subset of and of , that is, ∩ ⊆ and ∩ ⊆ 2. An element belongs to the union ∪ if belongs to or belongs to , hence, every element in belongs to ∪ , and every element in belongs to ∪. That is, ⊆ ∪ and ⊆ ∪ Set Theory | 24 DEPARTMENT of STATISTIC S By transitive property of subset, we can state the previous properties as:  Theorem 1.3 For any sets and , we have ∩ ⊆ ⊆ ∪ and ∩ ⊆ ⊆ ∪ Theorem 1.4  The following are equivalent: ⊆ , ∩ = , ∪ = Set Theory | 25 DEPARTMENT of STATISTICS Example: Ω = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} = {1, 3, 5, 6, 7, 9, 10} 3. Absolute Complement Then, = {2, 4, 8} The absolute complement or simply, complement of a set , Set Operations denoted by , is the set of elements which belongs to Ω but which do not belong to. ′ or ഥ  = ∈ , ∉ } Set Theory | 26 DEPARTMENT of STATISTICS simply the difference between and , denoted by 4. Difference \ or − , is the set The relative complement of a set of elements which belongs to with respect to set or, but which do not belong to.  \ = ∈ , ∉ } Example: = {1, 2, 3, 4, 5} = {2, 4, 6, 8} Then, \ = {1, 3, 5} and \ = 6, 8 Set Operations  Set Theory | 27 DEPARTMENT of STATISTICS ⊕ = {1, 3, 5, 6, 8} Set Operations 5. Symmetric Difference The symmetric difference of the sets and , denoted by ⊕ , consists of those elements which belong to or , but not in both. Example: = {1, 2, 3, 4, 5} ⊕ = ∪ \ ∩ or = {2, 4, 6, 8} ⊕ = \ ∪ ( \ ) Then,  Set Theory | 28 DEPARTMENT of S T A T I S T ICS on Set Operations 1.1.6 Laws Theorem 1.5. Idempotent Laws ∪ = ∩ = Theorem 1.6. Associative Laws ∪ ∪ = ∪ ( ∪ ) ∩ ∩ = ∩ ( ∩ ) Set Theory | 29 DEPARTMENT of S T A T I S T I C S Operations Laws on Set Theorem 1.7. Commutative Laws ∪ = ∪ ∩ = ∩ Theorem 1.8. Distributive Laws  ∪ ∩ = ∪ ∩ ( ∪ ) ∩ ∪ = ∩ ∪ ( ∩ ) Set Theory | 30 DEPARTMENT of STATISTICS Laws on Set Operations Theorem 1.9. Identity Laws ∪ Ø = ∩ = ∪ = ∩ Ø = Ø Theorem 1.10. Involution Law = Set Theory | 31 DEPARTMENT of S T A T I S T I C S Operations Laws on Set Theorem 1.11. Complement Laws ∪ = ∩ =Ø =ØØ = Theorem 1.12. De Morgan’s Laws  ∪ = ∩ ∩ = ∪ Set Theory | 32 DEPARTMENT of STATISTIC S Duality It can be observed that the identities on the Laws of Set Operations are arranged in pairs. It is a fact of set algebra, called the principle of duality, that, if  any equation is an identity, then its dual * is also an identity. The dual * of is the equation obtained by replacing each occurrence of ∪, ∩, Ω, and Ø in by ∩, ∪, Ø, and Ω, respectively. Set Theory | 33 DEPARTMENT of S T A T I S T I C S Elements in Finite Sets 1.1.7 Counting Lemma 1.13. Suppose and are finite disjoint sets. Then ∪ is finite and ∪ = + ( ) Lemma 1.14. Suppose is the disjoint union of finite sets and. Then is finite and = + ( ) Set Theory | 34 DEPARTMENT of STATISTIC S Consider, \ and ∩ are disjoint sets \ = − ∩ \ ∩ whose union is the set , that is, = \ ∪ ∩ Corollary 1.15. From Lemma 1.13,  ∩ = ( \ = \ + ) ∩ − Set Theory | 35 DEPARTMENT of STATISTIC S Consider also, Clearly, and are disjoint sets and whose union is Ω, that is  Ω = ∪ Ω − = Ω ( ) From Lemma 1.14, Corollary 1.16. Ω = + = − Set Theory | 36 DEPARTMENT of  STATISTIC S There is also a formula for ( ∪ ), even when the sets are not disjoint, called the Inclusion-Exclusion Principle. Theorem 1.17. Inclusion-Exclusion Principle ∪ = + − ( ∩ ) Illustration: = {1, 2, 3, 4, 5} = 2, 4, 6, 8  We have, ∪ = 1, 2, 3, 4, 5, 6, 8 and ∩ = {2, 4} ∪ = + − ( ∩ ) ∪ = 5 + 4 − 2 = 7 Set Theory | 37 DEPARTMENT of STATISTIC S Product Sets Consider two arbitrary sets and , the set of all ordered pair ( , ) where ∈ and ∈ is called the product or Cartesian product of and , denoted by  ×. By definition, × = , ∈ , ∈ } The Cartesian product deals with ordered pairs, so naturally the order in which the sets are considered is important. Set Theory | 38 DEPARTMENT of STATISTIC S Example: Let = { , , } and = 1, 2 , then  × = , 1 , , 2 , , 1 , , 2 , , 1 , ( , 2) and × = 1, , 1, , 1, , 2, , 2, , (2, ) Theorem 1.18. Suppose and are finite sets. Then × is finite and × = ⋅ Illustration: Using the previous example,  × = ⋅ × = 3 ⋅ 2 = 6 Set Theory | 39 DEPARTMENT of S T A T I S T I C S Partitions 1.1.8 Classes of Sets, Power Sets, Given a set Ω, we may wish to talk about some of its subsets. Class of Sets is a collection (or family) of sets. Illustration: Consider = 1, 2, 3 , some of its subsets are Ø, 1 , 2 , 2, 3 , and {1, 2}. When we get the collection of these subsets of , that will be Ø, 1 , 2 , 2, 3 ,{1, 2}.  Class of Set D Set Theory | 40 DEPARTMENT of STATISTIC S Power Set For a given set Ω, the power set of Ω is the set of all subsets of Ω and it will be denoted by (Ω).  If Ω is finite, then (Ω) is also finite. In fact, the ( ) cardinality is = Ø and Ω belong to (Ω) since they are subsets of Ω. Set Theory | 41 DEPARTMENT of STATISTIC S Example: Power Sets 1. Consider = 1, 2, 3  ( ) 3 = 2 =2 =8 = Ø, , 1 , 2 , 3 , 1, 2 , 1, 3 ,{2, 3} 2. For = , , 1, 2 ( ) 4 = 2 = 2 = 16 = Ø, , , , 1 , 2 , , , , 1 , , 2 ,{ , 1 , , 2 , 1, 2 , , , 1 , , 1, 2 , , , 2 ,{ , 1, 2}} Set Theory | 42 DEPARTMENT of STATISTIC S Partitions Let Ω be a non-empty set. A partition of Ω is a subdivision of Ω into non-overlapping, non-empty subsets A partition of Ω is a collection { } of non-empty subsets of Ω such that ✓ Each ∈ Ω belongs to one of the. ✓ The set of { } are mutually disjoint; that is, if ≠ , then ∩ = Ø The subsets in a partition are called cells.  Set Theory | 43 DEPARTMENT of STATISTIC S 1 = 1 Example: Partition 2 = 2, 3, 6 12 Consider Ω = {1, 2, 3, = {5, 4} 3 3 4, 5, 6}. 4 1, 2, and 3 One possible partition are cells. 5 of Ω is { 1, 2, 3} where: 6  Set Theory | 44 DEPARTMENT of STATISTIC S Indexed Classes of Sets An indexed class of sets, usually presented in the form | ∈ or  simply { } means that there is a set assigned to each ∈. The set is called the indexing set and the sets are said to be indexed by. The union of sets ∋ ‫ ڂ‬written ,, consists of those elements which belong to at least one of the. The intersection of sets ∋ ‫ځ‬ written ,, consists of those elements which belong to every. Set Theory | 45 DEPARTMENT of STATISTIC S Indexed Classes of Sets When the indexing set is the set ℕ of positive integers, the indexed class { 1, 2, 3, … } is called a sequence of sets. ∞ ራ =1 1 ∪ 2 ∞ ሩ =1 ∪ 3 ∪ ⋯ =  = ∩ 3 ∩ ⋯ 1 ∩ 2 Set Theory | 46 DEPARTMENT of S T A T I S T I CS Induction 1.1.9 Mathematical It is a standard procedure for establishing the validity of mathematical propositions involving a series of positive integers. Recall the steps in proving using Mathematical Induction Step 1: Check if the proposition is true for = 1. Step 2: Assume that the proposition is true for some positive integer =. Step 3: Prove that the proposition is also true for = + 1. Step 4: Make a conclusion. Set Theory | 47