Welcome To NAMASTE LECTURE SERIES

Welcome To NAMASTE LECTURE SERIES Set Theory Lecture # 1 Set UnionandIntersection 2009

Prof. Dr. Ram M. Shreshtha NAMASTE gd:]t HjHjnkf GOOD MORNING

At a Glance NAMASTE'S NEW NEPAL MATHS CENTRE Presents Set and Set Operations

Learning Objectives After completing this lesson, students should be able: • To describe a set using two standard forms of notation. • To perform operations on sets. • To use Venn diagrams to illustrate and solve problems involving sets

In the universe we live, we have our Solar System. Our Earth is in or belongs to the Solar System. The living beings in the earth is divided into - Plant Kingdom and Animal Kingdom. The animal kingdom contains two Families - vertebrate and Invertebrate. The huge human population is contained in the vertebrate family. In particular, the population of a country generally consists of different Social Classes. In every social class live people of different Income Groups. A Sample from every such income group may have some bright students. A bright student often gets high scores or Aggregate of marks in the examination. A student with high marks is often greeted with a Bunch of flowers.

(Continued) Words such as Aggregate, Bunch, Class, Family, Group, Kingdom, Population, Sample, System, etc. used above in different context indicate that they all convey something common. This idea or sense of commonness can be found in phrases such as collection of books in a library or deck of cards or ensemble of points also. Mathematicians use the simple word SET to convey the common idea contained in the words aggregate, bunch, class, deck, ensemble, family, kingdom, population, sample, system, etc.. This word set is taken as a fundamental term (without making any attempt to define it) to define other mathematical terms. This lesson is a brief introduction to the language and concepts involving sets, parts of sets, combination of sets, etc.

Part One An Animated Illustration

AGGREGATE COLLECTION BUNCH SET DECK FAMILY ENSEMBLE GROUP

The Solar System { } Sun Mercury Venus Earth MarsJupiter Saturn Uranus Neptune Basic Concepts Set and Set-membership Sun is inorbelongs toSolar system  Sun Solar system Earth is inor belongs toSolar system Neptuneis inor belongs toSolar system NeptuneSolar system EarthSolar system

The Solar System Sun

The Solar System Sub-Systems Star, Planets {Sun} Singleton set Star Sun {Sun}Solar System {Planets}Solar System Planets {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}

The Solar System {Sun} is a subset of Solar System {Sun}  Solar System {Inner Planets} is a subset of Solar System  Solar System {Inner planets} Mars Sun Inner planets Superior planets Outer planets {Outer planets}is a subset ofSolar System  {Outer planets} Solar System

The Solar System {Sun}  Solar System {Inner planets}  Solar System Solar System {Outer planets}   Solar System {Mars} Sun Inner planets Superior planets Outer planets Superior planets  Solar System

The Solar System (Sub-systems) {Sun}  Solar system Inner Planets  Solar system Outer Planets  Solar system Sun Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune {Outer planets } O {Sun} {Inner planets} I Basic Concepts : Set and Subset

The Solar System {Superior planets} {Outer planets} {Mars}  = {Inner planets} {Inner planets} {Mars}  = {Inner planets} Solar System  {Sun}  {Outer planets} = Sun Inner planets Superior planets Outer planets Basic Concept : Union of Sets

{Mars} Mars JupiterSaturnUranusNeptunePluto Set Union {Outer Planets} {Mars} {Outer Planets} = {Superior Planets}

{Inner Planets} MercuryVenusEarthMars JupiterSaturnUranusNeptunePluto Set Union I  O = P {Outer Planets}

The Solar System (Sub-systems) {Sun}  Solar system {Inner Planets}  Solar system {Outer Planets}  Solar system Sun Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune {Outer planets} O {Sun} {Inner planets} I {Mars} {Outer Planets} = { Superior Planets } {Sun}{Inner Planets}{Outer Planets}=Solar System Basic Concepts : Set , Subset and Union

The Solar System { } {Outer planets} {Mars} =  = Ø (Disjoint sets or sets with no common element) {Inner planets} {Mars} {Mars}  = (Intersecting sets or Sets with common element) {Inner planets} {Outer planets} Ø {Sun}   = Inner planets Sun Superior planets Outer planets (Intersecting sets or Sets with common element) {Inner planets} {Superior planets} {Mars}  =

{Inner Planets} SETINTERSECT ION MercuryVenusEarthMars MarsJupiterSaturnUranusNeptunePluto I  S = {Mars} {SuperiorPlanets}

The Solar System (Sub-systems) {Sun} {nner Planets}= {Inner Planets}  {Superior Planets}={ Mars} {Inner Planets}  {Outer Planets}= SunMercury Venus Earth MarsJupiter Saturn Uranus Neptune {Outer planets} O {Sun} {Inner planets} I Basic Concepts(Empty set, intersecting and disjoint sets} {Superior planets} S

The Solar System Complement of {Sun} = {Planets} {Mercury,Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune} Sun Complement of { Planets} = {Sun}

The Solar System { Sun}Inner PlanetsOuter Planets=Solar System Inner PlanetsOuter Planets= Outer PlanetsSuperior Planets Superior PlanetsInner Planets={Mars} Sun Inner planets Superior planets Outer planets

Part Two FUNDAMENTALS OF SET THEORY

A Bitof History Modern mathematics begins with the notion of a set. It lies at thefoundation of almost all branches of mathematics. Georg Cantor (1845 – 1915) is known as the father of the theory of sets.George Boole and John Venn were the two other 19th century mathematicians who made valuable contributions to the theory of sets.

Pioneers of Set Theory

Set The word set is synonymous with words such as: "aggregate", " bunch", “collection”, "dump", "ensemble", “family", " group" etc.Examples:The aggregate of marks; A bunch of flowers; A collection of stamps; A dump of waste materials;An ensemble of points; A family of plants;A group of students

Set notation A set is usually denoted by capital letters such asA, B, C, …, X, Y, Z;and the elements or members of a set by the small letters such asa, b, c, …, x, y, z.

Set Membership Set membership is indicated bythe symbol "" and non-membership is denoted by "".We writex X to mean “ x is an element or member of X”x A to mean “ x does not belong to A”

Set Specification a) By description/extension: It is the explicit listing of the objects or members or elements in between two braces or curly brackets { }. Examples: 1. A = {a, b, …, y, z}, 2. N = {1, 2, …, 11, 12 …}

Set Specification(Continued) b) By comprehension/ intension: It is the specification by a membership condition or rule for inclusion in the set. Examples: A = {x : x satisfies a property P(x)} = {x : x P(x)} = {x : P(x)}. This is theset-builder form. Here, the braces {x} is read as “The set of all x’s” the colon “:” as “ such that”.V={The set of vowels in the English alphabet}

Some Special Characteristics A set is basically unordered.It consists of distinct(or unequal) objects; Multiple listing of elements is not done.A set may consists of sets.A set may have nothing else or may be empty.A set with one element is called a singleton set. A set may have a fixed (orfinite) number of elements.

Successor Element In a non-empty non-singleton set, every element (except the last, if there is one such element) may be followed by a unique element, called the successor of the former. Examples:In the set of counting numbers2 is the successor of 13 is the successor of 2and so on.

Ordered Pair In a pair of numbers (i.e., a set with two elements),one number may be followed by another or one may be the successor of the other. Such a set is said to be ordered. An ordered pair in which bis a successor of a is denoted by (a, b).Two ordered pairs (a, b) and (c, d) are said to be equal if and only if a = c and b = d.

The Two Speical Cases a) Empty set. A set without any element is known as an empty set. In symbols,  = { x | x ≠ x }.Examples:1.The set of male-students in a girls’ school.2. The set of numbers in a fish pond.An important convention is to acceptthe empty set or null set or void set as a finite set

b) Universal set. The set of all elements under consideration. It is usually denoted by the letter U.The set of numbers is the universal set for ordinary arithmetical operations.

Venn Diagrams A Venn-diagram is the representation of the elements of a set by points inside a simple closed curve such as a circle or ellipse.The universal set U is usually denoted by a rectangle and its subset by a simple closed curve within the rectangle as follows:

a . e . i . o . . u Venn-Euler Diagram b. c. d . . . . . V . x . y . z U

Subset A set A is a subset of a set B if every element of A is an element.In symbols, we writeA  B if x  A then x  B.Examples:The set of boy-students in a class is a subset of the set of all students

2. a) T S 0 1 23 4 0 1 2 2. b) T S 0 1 2 0 1 2 S = T

Cases such as 2(a) and 2(b), when taken together, are denoted by the single notation: 2.The set of point in the circumference of a circle is a subset of the set of points of the circular region including the boundary Here T is called thesupersetofS.

Equal set Two sets are said to be equal if every element is also and element of the other. In other words, A set A is said to be equal to a set B if A is a subset of B and B is a subset of A if every element of A is an element of B. In symbols, we write A = B iff A  B and B  A.

Examples: • The set {1, 2, 3} = {3, 2, 1}; • but {123}≠ {321} 2. A B p p B A =

Proper Subset A subset A of a set B is a proper subset if the sets are not equal. A  B = { x | if x A then x B } and A ≠ B.Examples:1. The set {1, 5, 9} is a proper subset of the set {1, 2, 3, 4, 5, 6, 7, 8, 9}.2. The set of points on a line is a proper subset of the set of points in a plane.

Proper subset  B A c c B A

V e n n J o h n Diagram1923 Venn1834 Primes <10 Even integers from 2 to 9 2 0 4 6 8 1 -1 3 5 7 9 Positive integers less than 10 Odd integers from 1 to 9 Integers from -1 to 9

Power Set The power set of a set X is the set of all subsets of the set X. The power set contains both the empty set and the set X. In symbols, we write℘(X)= = 2X = { A | if x  A then x  X }.Example:The power set of S = {1, 2, 3} is the set℘(S)=2S ={,{1},{2},{3},{1,2},{1,3},{2,3},{1,2},{1,2,3}}

Equivalent Sets Two sets A and B are said to be equivalent if to each element of A there corresponds one element of B and to each element of B there corresponds one element of A ( i.e., if there is a one-to-one correspondence between the two sets).In symbols, we writeA ≈ Bto mean A is equivalent to B.

Examples: The set N of countries {Nepal, China, India} and the set C of capitals {Kathmandu, Beijing, Delhi} are equivalent as shown below: Nepal ChinaIndia ↕ ↕ ↕ KathmanduBeijingDelhi

1, 2, 3, … , 101, 102,… 2, 4, 6, … , 202, 204,… One-to-one correspondence

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