Sets

Awais Bakshy
Awais BakshyAwais Bakshy
Mathematics.
Chapter: Sets
Awais Bakshy
Sets.
 Set.
 A set is a collection of well-defined and distinct objects.
 Well-defined collection of distinct objects.
 Well-defined is means a collection which is such that, given any object, we
may be able to decide whether object belongs to the collection or not.
 Distinct objects means objects no two of which are identical (same)
 Set is denoted by capital letters. A, B, C, D, E, etc.
 Elements or members of set.
 The objects in the set are called elements or members of set.
 The elements or members of sets are In small letters.
Sets.
 Representation of a set .
 A set can be represented by three methods.
 Descriptive Method.
 A set can be described in words.
 For Example.
 The set of whole numbers.
 Tabular Method.
 A set can be described by listing its elements within brackets.
 For example.
 A={1, 2, 3, 4} B={3, 5, 7, 8,}
 Set Builder Method.
 A set can be described by using symbols and letters.
 For example.
 A={x/x ϵ N }
Sets.
 Some Sets.
 Integers.
 Positive Integers + 0 (Zero) + Negative Integers.
 Denoted By Capital Z.
 Z= {0, ±1, ±2, ±3, ±4, ±5, ±6, ±7, …}
 Negative Integers.
 Denoted by 𝑍′
(Z Dash)
 𝑍′
= {-1,-2, -3, -4, -5, -6, …}
 Natural Numbers.
 N ={1, 2, 3, 4, 5, 6, 7, …}
 Whole Numbers.
 W = {0, 1, 2, 3, 4, 5, 6, 7,8, …}
Sets.
 Even Numbers.
 E= {0, 2, 4, 6, 8, 10, 12, 14, …}
 Odd Number.
 O ={1, 3, 5, 7, 9, 11, 13, …}
 Real Numbers.
 R= Q+ 𝑄′
 Irrational Numbers.
 𝑄′
={ 3, 5, 11, 7, … }
 Rational Numbers.
 Q= {
𝑝
𝑞
| p, q ϵ Z }
Sets.
 Kinds of sets.
 Finite Set.
 A set in which all the members can be listed is called a finite set.
 A set which has limited members.
 For example.
 A={1, 2,3,4,5,6} B={3,5,7,9} C={0,1,2,3,4,5,6} D={2,4,8,10}
 Infinite Set.
 In some cases it is impossible to list all the members of a set. Such sets are
called infinite sets.
 A set which has unlimited members is called infinite set.
 For example.
 Z= {0, ±1, ±2, ±3, ±4, ±5, ±6, ±7, …}
 W = {0, 1, 2, 3, 4, 5, 6, 7,8, …}
 N ={1, 2, 3, 4, 5, 6, 7, …}
 O ={1, 3, 5, 7, 9, 11, 13, …}
Sets.
 Subset.
 When each member of set A is also a member of set B, then A is a subset of B.
 It is denoted by A ⊆ B (A is subset of B)
 For Example.
Let A={1, 2, 3,} B={1, 2, 3, 4,}
A ⊆ B and B ⊇ A
 Types of subset.
 There are two types of Subset.
 Proper subset.
 If A is subset of B and B contains at least one member which is not a member of
A, then A is Said to be proper subset of B.
 It is Denoted by A ⊂ B ( A is proper subset of B).
 For Example.
Let A={1, 2, 3,} B={1, 2, 3, 4,}
A ⊂ B
Sets.
 Improper subset.
 If A is subset of B and A=B, then we say that A is an improper subset of B . From
this definition it also follows that every set A is an improper subset of itself.
 For example.
Let A={a, b, c} B={c, a, b} and C={a, b, c, d}
A ⊂ C , B ⊂ C But A=B and B=A
 Equal Set.
 Two sets A and B are equal i.e., A=B “iff” they have the same elements that’s is,
if and only if every element of each set is an element of the other set.
 It is denoted by = (Equal) “iff”(if and only if)
 For Example.
Let. A={1,2,3} B={2,1,3}
A=b
Sets.
 Eqiulent set.
 If the elements of two sets A and B can be paired in such a way that each element of A is paired
with one and only one element of B and vice versa, then such a pairing is called a one to one
correspondence “ between A and B”.
 For example.
 If A={Bilal, Yasir, Ashfaq} and B={Fatima, Ummara, Samina} then six different (1-1)
correspondence can be established between A and B. Two of these correspondence are given
below .
i. {Bilal, Yasir, Ashfaq}
{Fatima, Ummara, Samina}
i. ii. {Bilal, Yasir, Ashfaq}
{Fatima, Ummara, Samina}
Sets.
 Two sets are said to be Eqiulent if (1-1) correspondence can be established
between them.
 The symbol ~ is used to mean is equivalent to , Thus A~B.
 Singleton Set.
 A set having only one element is called a singleton.
 For example.
A={5} B={8} C={0}
 Empty Set and Null Set.
 A set which contains no elements is called an empty set ( or null set).
 An empty set is a subset of ant set.
 It is denoted by { } , ∅
 For example.
A={ } B={∅} C=∅
Sets.
 Power set.
 A set may contain elements, which are sets themselves. For example if: S= set of
classes of a certain school, then elements of C are sets themselves because each
class is a set of students.
 The power set of set S denoted by P(S) is the set containing all the possible
subsets of S.
 For Example.
A={1, 2, 3}, then
P(S)={∅,{1}, {2},{3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }
 The power set of the empty set is not empty.
 2 𝑚 is formula for finding numbers elements in the power set.
 Universal Set.
 The set which contain all the members/elements in the discussion is known as
the universal set.
 Denoted by U.
Sets.
 Union of two sets.
 The union of two sets A and B is denoted by A ∪ B (A union B), is the set of all
elements, which belongs to A or B.
 Symbolically.
A ∪ B={x/x ∈ A ∨ x ∈ B }
 For example.
If A={1,2} and B={7,8} then.
A ∪ B={1,2,7,8}
 Intersection of Two sets.
 The intersection of two set A and B is the of elements which are common to both
A and B.
 It is denoted by A ∩ B (A intersection of B).
 For Example.
If A={1, 2, 3,4, 5, 6} and B={2, 4, 8, 10} then,
A ∩ B={2, 4}
Sets.
 Disjoint Set.
 If the intersection of two sets is the empty set then the sets are said to be
disjoint sets.
 For example.
 If A= Set to odd numbers and B= set of even numbers , then A and B are disjoint
sets.
 Overlapping Set.
 If intersection to two sets is non empty set is called overlapping set.
 For example.
If A={2,3,7,8} and B={3,5,7,9}, then
A ∩ B={3,7}
 Complement of a Set.
 Given the universal set U and a set A, then Complement of set A (denoted By 𝐴′
or 𝐴 𝑐
) is a set containing all the elements in U that are not in A.
 Complement of a set = A+𝐴′
= U
Sets.
 For example.
 If U={2,3,5,7,11,13} and A={2,3,7,13} , then
𝐴′= U -A={5,7}
 Difference two a set.
 The difference set of two sets A and B denoted by A-B consists of all the
elements which belongs to A but do not belong to B.
 The difference set of two sets B and A denoted by B-A consists of all the
elements which belongs to B but do not belong to A.
 Symbolically.
 A-B={x/x∈ A ∧ x ∉ B } and B-A={x/x∈ B ∧ x ∉ A }
Sets.
 For example.
 If A={1,2,3,4,5} and B={4,5,6,7,8,9,10} , then find A-B and B-A.
 Solution.
A-B=?
A-B={1,2,3,4,5}-{4,5,6,7,8,9,10}
A-B={1,2,3}
B-A=?
B-A={4,5,6,7,8,9,10}-{1,2,3,4,5}
B-A={6,7,8,9,10}.
 Note.
 A-B≠B-A.
Sets.
 Properties of Union and intersection.
 Commutative property of union.
 Example.
If A={1,2,3,4} and B={3,4,5,6,7,8} then prove A ∪ B =B ∪ A.
Solution.
L.H.S= A ∪ B
A ∪ B={1,2,3,4} ∪ {3,4,5,6,7,8}
A ∪ B={1,2,3,4,5,6,7,8}
R.H.S= B ∪ A
B ∪ A={3,4,5,6,7,8} ∪ {1,2,3,4}
B ∪ A={1,2,3,4,5,6,7,8}
Hence it is proved that A ∪ B =B ∪ A so it holds commutative property of union.
Sets.
 Properties of Union and intersection.
 Commutative property of intersection.
 Example.
If A={1,3,4} and B={3,4,5,6,7,} then prove A ∩ B = B ∩ A.
Solution.
L.H.S= A ∩ B
A ∩ B ={1,3,4} ∪ {3,4,5,6,7}
A ∩ B ={3,4,}
R.H.S= B ∩ A
B ∩ A ={3,4,5,6,7} ∪ {1,3,4}
B ∩ A ={3,4}
Hence it is proved that A ∩ B = B ∩ A so it holds commutative property of
intersection.
Sets.
 Associative property of union.
 Example.
If A={1,2,3,4} , B={3,4,5,6,7,8} and C={7,8,9} then prove A ∪ ( B ∪ C)=(A ∪ B) ∪ C.
Solution.
L.H.S= A ∪ ( B ∪ C)
B ∪ C=?
B ∪ C ={3,4,5,6,7,8} ∪ {7,8,9}
B ∪ C ={3,4,5,6,7,8,9}
A ∪ ( B ∪ C)={1,2,3,4} ∪ {3,4,5,6,7,8,9}
A ∪ ( B ∪ C) ={1,2,3,4,5,6,7,8.9}
R.H.S=(A ∪ B) ∪ C
A ∪ B=?
A ∪ B={1,2,3,4} ∪ {3,4,5,6,7,8}
A ∪ B={1,2,3,4,5,6,7,8}
(A ∪ B) ∪ C={1,2,3,4,5,6,7,8} ∪{7,8,9}
(A ∪ B) ∪ C={1,2,3,4,5,6,7,8,9}
Hence it is proved that A ∪ ( B ∪ C)=(A ∪ B) ∪ C so it holds Associative property of union.
Sets.
 Associative property of intersection.
 Example.
If A={1,2,3,4} , B={3,4,5,6,7,8} and C={4,7,8,9} then prove A ∩( B ∩ C)=(A ∩ B) ∩ C.
Solution.
L.H.S= A ∩ ( B ∩ C)
B ∩ C=?
B ∩ C ={3,4,5,6,7,8} ∩ {4,7,8,9}
B ∩ C ={4,7,8}
A ∩ ( B ∩ C)={1,2,3,4} ∩ {4,7,8}
A ∩ ( B ∩ C) ={4}
R.H.S=(A ∩ B) ∩ C
A ∩ B=?
A ∩ B={1,2,3,4} ∩ {3,4,5,6,7,8}
A ∩ B={3,4}
(A ∩ B) ∩ C={3,4} ∩{4,7,8,9}
(A ∩ B) ∩ C={4}
Hence it is proved that A ∩ ( B ∩ C)=(A ∩ B) ∩ C so it holds Associative property of intersection
Sets.
 Distributive property of union over intersection.
 Example.
If A={1,2,3,4} , B={3,4,5,7,8} and C={4,5,7,8,9} then prove A ∪( B ∩ C) = (A ∪ B) ∩ (A ∪ C).
Solution.
L.H.S= A ∪( B ∩ C)
B ∩ C=?
B ∩ C ={3,4,5,7,8} ∩ {4,5,7,8,9}
B ∩ C ={4,5,7,8}
A ∪ ( B ∩ C)={1,2,3,4} ∪ {4,5,7,8}
A ∪ ( B ∩ C) ={1,2,3,4,5,7,8}
R.H.S = (A ∪ B) ∩ (A ∪ C)
A ∪ B=?
A ∪ B={1,2,3,4} ∪ {3,4,5,7,8}
A ∪ B={1,2,3,4,5,7,8}
A ∪ C=?
A ∪ C={1,2,3,4} ∪ {4,5,7,8,9}
A ∪ C={1,2,3,4,5,7,8,9}
(A ∪ B) ∩ (A ∪ C) = {1,2,3,4,5,7,8} ∩ {1,2,3,4,5,7,8,9}
(A ∪ B) ∩ (A ∪ C) = {1,2,3,4,5,,7,8}
Hence it is proved that A ∪( B ∩ C) = (A ∪ B) ∩ (A ∪ C) so it holds distributive property of union over intersection .
Sets.
 Distributive property of intersection over Union.
 Example.
If A={1,2,3,4} , B={3,4,5,7,8} and C={4,5,7,8,9} then prove A ∩( B ∪ C) = (A ∩ B) ∪ (A ∩ C).
Solution.
L.H.S= A ∩( B ∪ C)
B ∪ C=?
B ∪ C ={3,4,5,7,8} ∩ {4,5,7,8,9}
B ∪ C ={3,4,5,7,8,9}
A ∩ ( B ∪ C)={1,2,3,4} ∩ {3,4,5,7,8,9}
A ∩ ( B ∪ C)={3,4}
R.H.S = (A ∩ B) ∩ (A ∩ C)
A ∩ B=?
A ∩ B={1,2,3,4} ∩ {3,4,5,7,8}
A ∩ B={3,4}
A ∩ C=?
A ∩ C={1,2,3,4} ∩ {4,5,7,8,9}
A ∩ C={4}
(A ∩ B) ∩ (A ∩ C) = {3,4} ∩{4}
(A ∩ B) ∩ (A ∩ C) = {3,4}
Hence it is proved that A ∩( B ∪ C) = (A ∩ B) ∪ (A ∩ C) so it holds distributive property of intersection over union .
Sets.
 Exercise 1.
 Q N0 1: Prof all conditions of De Morgan's law is A={a,b,c,d,e,f,g,h} and B={a,b,c,e,g,i,j}
 (A ∪ B)′=𝐴′ ∩ 𝐵′
 (A ∩ B)′
=𝐴′
∪ 𝐵′
 Q No 2: Solve all properties of union and intersection If A={2,4,6,8,10} , B={1,2,3,4,5,6}
and C={0,1,2,3,4,5,6,7,8,9,10}
 A ∪ B =B ∪ A.
 A ∩ B = B ∩ A
 A ∪ ( B ∪ C)=(A ∪ B) ∪ C
 A ∩( B ∩ C)=(A ∩ B) ∩ C
 A ∪( B ∩ C) = (A ∪ B) ∩ (A ∪ C).
 A ∩( B ∪ C) = (A ∩ B) ∪ (A ∩ C)
1 de 22

Recomendados

Types Of SetTypes Of Set
Types Of SetPkwebbs
1.1K vistas31 diapositivas
SetsSets
SetsJohn Carl Carcero
2.3K vistas41 diapositivas
Matrix.Matrix.
Matrix.Awais Bakshy
8.4K vistas12 diapositivas
Sets PowerPoint PresentationSets PowerPoint Presentation
Sets PowerPoint PresentationAshna Rajput
2.5K vistas29 diapositivas

Más contenido relacionado

La actualidad más candente

Chapter   1, SetsChapter   1, Sets
Chapter 1, SetsAnanya Sharma
2.5K vistas19 diapositivas
Sets and venn diagramsSets and venn diagrams
Sets and venn diagramsFarhana Shaheen
1.5K vistas50 diapositivas
Set operationsSet operations
Set operationsrajshreemuthiah
1K vistas14 diapositivas
SetsSets
Setsnischayyy
5.4K vistas16 diapositivas
SET THEORYSET THEORY
SET THEORYLena
60K vistas58 diapositivas

La actualidad más candente(20)

Chapter   1, SetsChapter   1, Sets
Chapter 1, Sets
Ananya Sharma2.5K vistas
Sets and venn diagramsSets and venn diagrams
Sets and venn diagrams
Farhana Shaheen1.5K vistas
Set operationsSet operations
Set operations
rajshreemuthiah1K vistas
SetsSets
Sets
nischayyy5.4K vistas
SET THEORYSET THEORY
SET THEORY
Lena60K vistas
Set theorySet theory
Set theory
AN_Rajin5.2K vistas
Sets class 11Sets class 11
Sets class 11
Nitishkumar01059995.1K vistas
1. set theory1. set theory
1. set theory
caymulb202 vistas
LogarithmsLogarithms
Logarithms
Birinder Singh Gulati8.3K vistas
Introduction to SetsIntroduction to Sets
Introduction to Sets
Ashita Agrawal46.1K vistas
RingRing
Ring
Muhammad Umar Farooq7.5K vistas
Matrix algebraMatrix algebra
Matrix algebra
Farzad Javidanrad17.8K vistas
Maths sets pptMaths sets ppt
Maths sets ppt
Akshit Saxena116.1K vistas
Group  abstract algebraGroup  abstract algebra
Group abstract algebra
NaliniSPatil3.9K vistas
Matrix OperationsMatrix Operations
Matrix Operations
Ron Eick3.2K vistas
Module week 1 Q1Module week 1 Q1
Module week 1 Q1
Rommel Limbauan112 vistas
How to Find a Cartesian ProductHow to Find a Cartesian Product
How to Find a Cartesian Product
Don Sevcik2K vistas
LogarithmsLogarithms
Logarithms
siking2620K vistas
Basics of Counting TechniquesBasics of Counting Techniques
Basics of Counting Techniques
Efren Medallo5K vistas

Similar a Sets

Basic concepts on set.pdfBasic concepts on set.pdf
Basic concepts on set.pdfSmita828451
35 vistas4 diapositivas
mathematical sets.pdfmathematical sets.pdf
mathematical sets.pdfJihudumie.Com
21 vistas36 diapositivas
SET AND ITS OPERATIONSSET AND ITS OPERATIONS
SET AND ITS OPERATIONSRohithV15
214 vistas12 diapositivas

Similar a Sets(20)

POWERPOINT (SETS & FUNCTIONS).pdfPOWERPOINT (SETS & FUNCTIONS).pdf
POWERPOINT (SETS & FUNCTIONS).pdf
MaryAnnBatac1101 vistas
Basic concepts on set.pdfBasic concepts on set.pdf
Basic concepts on set.pdf
Smita82845135 vistas
Sets functions-sequences-exercisesSets functions-sequences-exercises
Sets functions-sequences-exercises
Roshayu Mohamad161 vistas
mathematical sets.pdfmathematical sets.pdf
mathematical sets.pdf
Jihudumie.Com21 vistas
CMSC 56 | Lecture 6: Sets & Set OperationsCMSC 56 | Lecture 6: Sets & Set Operations
CMSC 56 | Lecture 6: Sets & Set Operations
allyn joy calcaben1.1K vistas
SET AND ITS OPERATIONSSET AND ITS OPERATIONS
SET AND ITS OPERATIONS
RohithV15214 vistas
2 》set operation.pdf2 》set operation.pdf
2 》set operation.pdf
HamayonHelali12 vistas
Set theory-pptSet theory-ppt
Set theory-ppt
vipulAtri269 vistas
Sets and there different types.Sets and there different types.
Sets and there different types.
Ashufb232311.8K vistas
SETS-AND-SUBSETS.pptxSETS-AND-SUBSETS.pptx
SETS-AND-SUBSETS.pptx
JuanMiguelTangkeko13 vistas
Sets in Maths (Complete Topic)Sets in Maths (Complete Topic)
Sets in Maths (Complete Topic)
Manik Bhola862 vistas
Himpunan plpgHimpunan plpg
Himpunan plpg
Nom Mujib323 vistas
Sets in discrete mathematicsSets in discrete mathematics
Sets in discrete mathematics
University of Potsdam1.2K vistas
Discrete mathematics OR Structure Discrete mathematics OR Structure
Discrete mathematics OR Structure
Abdullah Jan445 vistas
schaums-probability.pdfschaums-probability.pdf
schaums-probability.pdf
Sahat Hutajulu313 vistas
Sets (Mathematics class XI)Sets (Mathematics class XI)
Sets (Mathematics class XI)
VihaanBhambhani834 vistas
4898850.ppt4898850.ppt
4898850.ppt
UsamaManzoorLucky120 vistas
Set conceptsSet concepts
Set concepts
AarjavPinara257 vistas
Set Theory - CSE101 CSE GUB BDSet Theory - CSE101 CSE GUB BD
Set Theory - CSE101 CSE GUB BD
Md. Shahidul Islam Prodhan35 vistas
2.1 Sets2.1 Sets
2.1 Sets
showslidedump10.4K vistas

Más de Awais Bakshy

Number systemNumber system
Number systemAwais Bakshy
177 vistas8 diapositivas
Function and Its Types.Function and Its Types.
Function and Its Types.Awais Bakshy
613 vistas11 diapositivas
Network Typologies.Network Typologies.
Network Typologies.Awais Bakshy
661 vistas14 diapositivas
Atmospheric pollutionAtmospheric pollution
Atmospheric pollutionAwais Bakshy
415 vistas58 diapositivas
Sedimentary rocks bs 1st yearSedimentary rocks bs 1st year
Sedimentary rocks bs 1st yearAwais Bakshy
740 vistas32 diapositivas
Metamorphic rocks bs 1st yearMetamorphic rocks bs 1st year
Metamorphic rocks bs 1st yearAwais Bakshy
3.9K vistas18 diapositivas

Más de Awais Bakshy(18)

Number systemNumber system
Number system
Awais Bakshy177 vistas
Function and Its Types.Function and Its Types.
Function and Its Types.
Awais Bakshy613 vistas
Network Typologies.Network Typologies.
Network Typologies.
Awais Bakshy661 vistas
Atmospheric pollutionAtmospheric pollution
Atmospheric pollution
Awais Bakshy415 vistas
Sedimentary rocks bs 1st yearSedimentary rocks bs 1st year
Sedimentary rocks bs 1st year
Awais Bakshy740 vistas
Metamorphic rocks bs 1st yearMetamorphic rocks bs 1st year
Metamorphic rocks bs 1st year
Awais Bakshy3.9K vistas
Igneous rocks  bs 1st yearIgneous rocks  bs 1st year
Igneous rocks bs 1st year
Awais Bakshy507 vistas
Plate tectonics bs 1st yearPlate tectonics bs 1st year
Plate tectonics bs 1st year
Awais Bakshy6.5K vistas
Lecture volcanoes.Lecture volcanoes.
Lecture volcanoes.
Awais Bakshy370 vistas
Earth's interior bs 1st yearEarth's interior bs 1st year
Earth's interior bs 1st year
Awais Bakshy884 vistas
Earthquakes.Earthquakes.
Earthquakes.
Awais Bakshy102 vistas
General introduction to geologyGeneral introduction to geology
General introduction to geology
Awais Bakshy225 vistas
Computer science.Computer science.
Computer science.
Awais Bakshy85 vistas
Minerals.Minerals.
Minerals.
Awais Bakshy522 vistas
Geography.Geography.
Geography.
Awais Bakshy240 vistas
Geological time scale.Geological time scale.
Geological time scale.
Awais Bakshy5.6K vistas

Último(20)

Plastic waste.pdfPlastic waste.pdf
Plastic waste.pdf
alqaseedae94 vistas
ACTIVITY BOOK key water sports.pptxACTIVITY BOOK key water sports.pptx
ACTIVITY BOOK key water sports.pptx
Mar Caston Palacio275 vistas
7 NOVEL DRUG DELIVERY SYSTEM.pptx7 NOVEL DRUG DELIVERY SYSTEM.pptx
7 NOVEL DRUG DELIVERY SYSTEM.pptx
Sachin Nitave56 vistas
SIMPLE PRESENT TENSE_new.pptxSIMPLE PRESENT TENSE_new.pptx
SIMPLE PRESENT TENSE_new.pptx
nisrinamadani2159 vistas
STERILITY TEST.pptxSTERILITY TEST.pptx
STERILITY TEST.pptx
Anupkumar Sharma107 vistas
Education and Diversity.pptxEducation and Diversity.pptx
Education and Diversity.pptx
DrHafizKosar87 vistas
Narration  ppt.pptxNarration  ppt.pptx
Narration ppt.pptx
TARIQ KHAN76 vistas
Universe revised.pdfUniverse revised.pdf
Universe revised.pdf
DrHafizKosar88 vistas
Use of Probiotics in Aquaculture.pptxUse of Probiotics in Aquaculture.pptx
Use of Probiotics in Aquaculture.pptx
AKSHAY MANDAL72 vistas
Google solution challenge..pptxGoogle solution challenge..pptx
Google solution challenge..pptx
ChitreshGyanani170 vistas
Chemistry of sex hormones.pptxChemistry of sex hormones.pptx
Chemistry of sex hormones.pptx
RAJ K. MAURYA107 vistas
Material del tarjetero LEES Travesías.docxMaterial del tarjetero LEES Travesías.docx
Material del tarjetero LEES Travesías.docx
Norberto Millán Muñoz60 vistas
ICS3211_lecture 08_2023.pdfICS3211_lecture 08_2023.pdf
ICS3211_lecture 08_2023.pdf
Vanessa Camilleri79 vistas
ANATOMY AND PHYSIOLOGY UNIT 1 { PART-1}ANATOMY AND PHYSIOLOGY UNIT 1 { PART-1}
ANATOMY AND PHYSIOLOGY UNIT 1 { PART-1}
DR .PALLAVI PATHANIA190 vistas
Student Voice Student Voice
Student Voice
Pooky Knightsmith125 vistas

Sets

  • 2. Sets.  Set.  A set is a collection of well-defined and distinct objects.  Well-defined collection of distinct objects.  Well-defined is means a collection which is such that, given any object, we may be able to decide whether object belongs to the collection or not.  Distinct objects means objects no two of which are identical (same)  Set is denoted by capital letters. A, B, C, D, E, etc.  Elements or members of set.  The objects in the set are called elements or members of set.  The elements or members of sets are In small letters.
  • 3. Sets.  Representation of a set .  A set can be represented by three methods.  Descriptive Method.  A set can be described in words.  For Example.  The set of whole numbers.  Tabular Method.  A set can be described by listing its elements within brackets.  For example.  A={1, 2, 3, 4} B={3, 5, 7, 8,}  Set Builder Method.  A set can be described by using symbols and letters.  For example.  A={x/x ϵ N }
  • 4. Sets.  Some Sets.  Integers.  Positive Integers + 0 (Zero) + Negative Integers.  Denoted By Capital Z.  Z= {0, ±1, ±2, ±3, ±4, ±5, ±6, ±7, …}  Negative Integers.  Denoted by 𝑍′ (Z Dash)  𝑍′ = {-1,-2, -3, -4, -5, -6, …}  Natural Numbers.  N ={1, 2, 3, 4, 5, 6, 7, …}  Whole Numbers.  W = {0, 1, 2, 3, 4, 5, 6, 7,8, …}
  • 5. Sets.  Even Numbers.  E= {0, 2, 4, 6, 8, 10, 12, 14, …}  Odd Number.  O ={1, 3, 5, 7, 9, 11, 13, …}  Real Numbers.  R= Q+ 𝑄′  Irrational Numbers.  𝑄′ ={ 3, 5, 11, 7, … }  Rational Numbers.  Q= { 𝑝 𝑞 | p, q ϵ Z }
  • 6. Sets.  Kinds of sets.  Finite Set.  A set in which all the members can be listed is called a finite set.  A set which has limited members.  For example.  A={1, 2,3,4,5,6} B={3,5,7,9} C={0,1,2,3,4,5,6} D={2,4,8,10}  Infinite Set.  In some cases it is impossible to list all the members of a set. Such sets are called infinite sets.  A set which has unlimited members is called infinite set.  For example.  Z= {0, ±1, ±2, ±3, ±4, ±5, ±6, ±7, …}  W = {0, 1, 2, 3, 4, 5, 6, 7,8, …}  N ={1, 2, 3, 4, 5, 6, 7, …}  O ={1, 3, 5, 7, 9, 11, 13, …}
  • 7. Sets.  Subset.  When each member of set A is also a member of set B, then A is a subset of B.  It is denoted by A ⊆ B (A is subset of B)  For Example. Let A={1, 2, 3,} B={1, 2, 3, 4,} A ⊆ B and B ⊇ A  Types of subset.  There are two types of Subset.  Proper subset.  If A is subset of B and B contains at least one member which is not a member of A, then A is Said to be proper subset of B.  It is Denoted by A ⊂ B ( A is proper subset of B).  For Example. Let A={1, 2, 3,} B={1, 2, 3, 4,} A ⊂ B
  • 8. Sets.  Improper subset.  If A is subset of B and A=B, then we say that A is an improper subset of B . From this definition it also follows that every set A is an improper subset of itself.  For example. Let A={a, b, c} B={c, a, b} and C={a, b, c, d} A ⊂ C , B ⊂ C But A=B and B=A  Equal Set.  Two sets A and B are equal i.e., A=B “iff” they have the same elements that’s is, if and only if every element of each set is an element of the other set.  It is denoted by = (Equal) “iff”(if and only if)  For Example. Let. A={1,2,3} B={2,1,3} A=b
  • 9. Sets.  Eqiulent set.  If the elements of two sets A and B can be paired in such a way that each element of A is paired with one and only one element of B and vice versa, then such a pairing is called a one to one correspondence “ between A and B”.  For example.  If A={Bilal, Yasir, Ashfaq} and B={Fatima, Ummara, Samina} then six different (1-1) correspondence can be established between A and B. Two of these correspondence are given below . i. {Bilal, Yasir, Ashfaq} {Fatima, Ummara, Samina} i. ii. {Bilal, Yasir, Ashfaq} {Fatima, Ummara, Samina}
  • 10. Sets.  Two sets are said to be Eqiulent if (1-1) correspondence can be established between them.  The symbol ~ is used to mean is equivalent to , Thus A~B.  Singleton Set.  A set having only one element is called a singleton.  For example. A={5} B={8} C={0}  Empty Set and Null Set.  A set which contains no elements is called an empty set ( or null set).  An empty set is a subset of ant set.  It is denoted by { } , ∅  For example. A={ } B={∅} C=∅
  • 11. Sets.  Power set.  A set may contain elements, which are sets themselves. For example if: S= set of classes of a certain school, then elements of C are sets themselves because each class is a set of students.  The power set of set S denoted by P(S) is the set containing all the possible subsets of S.  For Example. A={1, 2, 3}, then P(S)={∅,{1}, {2},{3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }  The power set of the empty set is not empty.  2 𝑚 is formula for finding numbers elements in the power set.  Universal Set.  The set which contain all the members/elements in the discussion is known as the universal set.  Denoted by U.
  • 12. Sets.  Union of two sets.  The union of two sets A and B is denoted by A ∪ B (A union B), is the set of all elements, which belongs to A or B.  Symbolically. A ∪ B={x/x ∈ A ∨ x ∈ B }  For example. If A={1,2} and B={7,8} then. A ∪ B={1,2,7,8}  Intersection of Two sets.  The intersection of two set A and B is the of elements which are common to both A and B.  It is denoted by A ∩ B (A intersection of B).  For Example. If A={1, 2, 3,4, 5, 6} and B={2, 4, 8, 10} then, A ∩ B={2, 4}
  • 13. Sets.  Disjoint Set.  If the intersection of two sets is the empty set then the sets are said to be disjoint sets.  For example.  If A= Set to odd numbers and B= set of even numbers , then A and B are disjoint sets.  Overlapping Set.  If intersection to two sets is non empty set is called overlapping set.  For example. If A={2,3,7,8} and B={3,5,7,9}, then A ∩ B={3,7}  Complement of a Set.  Given the universal set U and a set A, then Complement of set A (denoted By 𝐴′ or 𝐴 𝑐 ) is a set containing all the elements in U that are not in A.  Complement of a set = A+𝐴′ = U
  • 14. Sets.  For example.  If U={2,3,5,7,11,13} and A={2,3,7,13} , then 𝐴′= U -A={5,7}  Difference two a set.  The difference set of two sets A and B denoted by A-B consists of all the elements which belongs to A but do not belong to B.  The difference set of two sets B and A denoted by B-A consists of all the elements which belongs to B but do not belong to A.  Symbolically.  A-B={x/x∈ A ∧ x ∉ B } and B-A={x/x∈ B ∧ x ∉ A }
  • 15. Sets.  For example.  If A={1,2,3,4,5} and B={4,5,6,7,8,9,10} , then find A-B and B-A.  Solution. A-B=? A-B={1,2,3,4,5}-{4,5,6,7,8,9,10} A-B={1,2,3} B-A=? B-A={4,5,6,7,8,9,10}-{1,2,3,4,5} B-A={6,7,8,9,10}.  Note.  A-B≠B-A.
  • 16. Sets.  Properties of Union and intersection.  Commutative property of union.  Example. If A={1,2,3,4} and B={3,4,5,6,7,8} then prove A ∪ B =B ∪ A. Solution. L.H.S= A ∪ B A ∪ B={1,2,3,4} ∪ {3,4,5,6,7,8} A ∪ B={1,2,3,4,5,6,7,8} R.H.S= B ∪ A B ∪ A={3,4,5,6,7,8} ∪ {1,2,3,4} B ∪ A={1,2,3,4,5,6,7,8} Hence it is proved that A ∪ B =B ∪ A so it holds commutative property of union.
  • 17. Sets.  Properties of Union and intersection.  Commutative property of intersection.  Example. If A={1,3,4} and B={3,4,5,6,7,} then prove A ∩ B = B ∩ A. Solution. L.H.S= A ∩ B A ∩ B ={1,3,4} ∪ {3,4,5,6,7} A ∩ B ={3,4,} R.H.S= B ∩ A B ∩ A ={3,4,5,6,7} ∪ {1,3,4} B ∩ A ={3,4} Hence it is proved that A ∩ B = B ∩ A so it holds commutative property of intersection.
  • 18. Sets.  Associative property of union.  Example. If A={1,2,3,4} , B={3,4,5,6,7,8} and C={7,8,9} then prove A ∪ ( B ∪ C)=(A ∪ B) ∪ C. Solution. L.H.S= A ∪ ( B ∪ C) B ∪ C=? B ∪ C ={3,4,5,6,7,8} ∪ {7,8,9} B ∪ C ={3,4,5,6,7,8,9} A ∪ ( B ∪ C)={1,2,3,4} ∪ {3,4,5,6,7,8,9} A ∪ ( B ∪ C) ={1,2,3,4,5,6,7,8.9} R.H.S=(A ∪ B) ∪ C A ∪ B=? A ∪ B={1,2,3,4} ∪ {3,4,5,6,7,8} A ∪ B={1,2,3,4,5,6,7,8} (A ∪ B) ∪ C={1,2,3,4,5,6,7,8} ∪{7,8,9} (A ∪ B) ∪ C={1,2,3,4,5,6,7,8,9} Hence it is proved that A ∪ ( B ∪ C)=(A ∪ B) ∪ C so it holds Associative property of union.
  • 19. Sets.  Associative property of intersection.  Example. If A={1,2,3,4} , B={3,4,5,6,7,8} and C={4,7,8,9} then prove A ∩( B ∩ C)=(A ∩ B) ∩ C. Solution. L.H.S= A ∩ ( B ∩ C) B ∩ C=? B ∩ C ={3,4,5,6,7,8} ∩ {4,7,8,9} B ∩ C ={4,7,8} A ∩ ( B ∩ C)={1,2,3,4} ∩ {4,7,8} A ∩ ( B ∩ C) ={4} R.H.S=(A ∩ B) ∩ C A ∩ B=? A ∩ B={1,2,3,4} ∩ {3,4,5,6,7,8} A ∩ B={3,4} (A ∩ B) ∩ C={3,4} ∩{4,7,8,9} (A ∩ B) ∩ C={4} Hence it is proved that A ∩ ( B ∩ C)=(A ∩ B) ∩ C so it holds Associative property of intersection
  • 20. Sets.  Distributive property of union over intersection.  Example. If A={1,2,3,4} , B={3,4,5,7,8} and C={4,5,7,8,9} then prove A ∪( B ∩ C) = (A ∪ B) ∩ (A ∪ C). Solution. L.H.S= A ∪( B ∩ C) B ∩ C=? B ∩ C ={3,4,5,7,8} ∩ {4,5,7,8,9} B ∩ C ={4,5,7,8} A ∪ ( B ∩ C)={1,2,3,4} ∪ {4,5,7,8} A ∪ ( B ∩ C) ={1,2,3,4,5,7,8} R.H.S = (A ∪ B) ∩ (A ∪ C) A ∪ B=? A ∪ B={1,2,3,4} ∪ {3,4,5,7,8} A ∪ B={1,2,3,4,5,7,8} A ∪ C=? A ∪ C={1,2,3,4} ∪ {4,5,7,8,9} A ∪ C={1,2,3,4,5,7,8,9} (A ∪ B) ∩ (A ∪ C) = {1,2,3,4,5,7,8} ∩ {1,2,3,4,5,7,8,9} (A ∪ B) ∩ (A ∪ C) = {1,2,3,4,5,,7,8} Hence it is proved that A ∪( B ∩ C) = (A ∪ B) ∩ (A ∪ C) so it holds distributive property of union over intersection .
  • 21. Sets.  Distributive property of intersection over Union.  Example. If A={1,2,3,4} , B={3,4,5,7,8} and C={4,5,7,8,9} then prove A ∩( B ∪ C) = (A ∩ B) ∪ (A ∩ C). Solution. L.H.S= A ∩( B ∪ C) B ∪ C=? B ∪ C ={3,4,5,7,8} ∩ {4,5,7,8,9} B ∪ C ={3,4,5,7,8,9} A ∩ ( B ∪ C)={1,2,3,4} ∩ {3,4,5,7,8,9} A ∩ ( B ∪ C)={3,4} R.H.S = (A ∩ B) ∩ (A ∩ C) A ∩ B=? A ∩ B={1,2,3,4} ∩ {3,4,5,7,8} A ∩ B={3,4} A ∩ C=? A ∩ C={1,2,3,4} ∩ {4,5,7,8,9} A ∩ C={4} (A ∩ B) ∩ (A ∩ C) = {3,4} ∩{4} (A ∩ B) ∩ (A ∩ C) = {3,4} Hence it is proved that A ∩( B ∪ C) = (A ∩ B) ∪ (A ∩ C) so it holds distributive property of intersection over union .
  • 22. Sets.  Exercise 1.  Q N0 1: Prof all conditions of De Morgan's law is A={a,b,c,d,e,f,g,h} and B={a,b,c,e,g,i,j}  (A ∪ B)′=𝐴′ ∩ 𝐵′  (A ∩ B)′ =𝐴′ ∪ 𝐵′  Q No 2: Solve all properties of union and intersection If A={2,4,6,8,10} , B={1,2,3,4,5,6} and C={0,1,2,3,4,5,6,7,8,9,10}  A ∪ B =B ∪ A.  A ∩ B = B ∩ A  A ∪ ( B ∪ C)=(A ∪ B) ∪ C  A ∩( B ∩ C)=(A ∩ B) ∩ C  A ∪( B ∩ C) = (A ∪ B) ∩ (A ∪ C).  A ∩( B ∪ C) = (A ∩ B) ∪ (A ∩ C)