# 2.1 Sets

1 of 27

## Recommended

Set, Relations and Functions
Set, Relations and Functionssuthi

Math 7 | lesson 1 Sets
Math 7 | lesson 1 SetsAriel Gilbuena

ppt for Properties of the Operations on Integers
ppt for Properties of the Operations on Integersneria_ayren

1. sets and basic notations
1. sets and basic notationseduardman

## What's hot

Solving Word Problems Involving Quadratic Equations
Solving Word Problems Involving Quadratic Equationskliegey524

Relations and Functions
Relations and Functionstoni dimella

Factoring Sum and Difference of Two Cubes
Factoring Sum and Difference of Two CubesFree Math Powerpoints

Addition and subtraction of rational expression
Addition and subtraction of rational expressionMartinGeraldine

PROPERTIES OF THE OPERATIONS ON INTEGERS
PROPERTIES OF THE OPERATIONS ON INTEGERSKheysho Bevz Alinsub

Finite and Infinite_Equal and Equivalent_Ways of naming sets
Finite and Infinite_Equal and Equivalent_Ways of naming setsFree Math Powerpoints

Set Concepts
Set Conceptsshbest

Rational functions
Rational functionszozima

sets and venn diagrams
sets and venn diagramsShahmir Ali

Solving Quadratic Equations by FactoringFree Math Powerpoints

Introduction to sets
Introduction to setsSonia Pahuja

Rational algebraic expressions
Rational algebraic expressionsmyla gambalan

Lesson 1.2 the set of real numbers
Lesson 1.2 the set of real numbersJohnnyBallecer

### What's hot(20)

Solving Word Problems Involving Quadratic Equations
Solving Word Problems Involving Quadratic Equations

Relations and Functions
Relations and Functions

Factoring Sum and Difference of Two Cubes
Factoring Sum and Difference of Two Cubes

Addition and subtraction of rational expression
Addition and subtraction of rational expression

PROPERTIES OF THE OPERATIONS ON INTEGERS
PROPERTIES OF THE OPERATIONS ON INTEGERS

Sets
Sets

Finite and Infinite_Equal and Equivalent_Ways of naming sets
Finite and Infinite_Equal and Equivalent_Ways of naming sets

Set Concepts
Set Concepts

Rational functions
Rational functions

sets and venn diagrams
sets and venn diagrams

Introduction to sets
Introduction to sets

Rational algebraic expressions
Rational algebraic expressions

Lesson 1.2 the set of real numbers
Lesson 1.2 the set of real numbers

Subsets of A Line
Subsets of A Line

## Viewers also liked

Sets and Functions By Saleh ElShehabey
Sets and Functions By Saleh ElShehabeyravingeek

Ppt sets and set operations
Ppt sets and set operationsgeckbanaag

Set Theory
Set Theoryitutor

Elements Of Plot
Elements Of Plotkjhatzi

Descriptive Research Method
Descriptive Research MethodRenu Sharma

MCQ's for 8th class All book mxq's before mide term
MCQ's for 8th class All book mxq's before mide termFazaia inter college lahore

SUBJECT AND PREDICATE
SUBJECT AND PREDICATELaarni Eguia

Subjects and predicates
Subjects and predicatesTeresa Wilson

Mathematics Sets and Logic Week 1
Mathematics Sets and Logic Week 1Jasmine Jesse

Maths Project on sets
Maths Project on setsatifansari17

Parts Of A Book
Parts Of A BookNWEMS

Mathematics class XI SETS
Mathematics class XI SETSNaveen R

Elements Of Plot Cinderella
Elements Of Plot Cinderellamrswjohnston

SET THEORY
SET THEORYLena

### Viewers also liked(20)

Maths sets ppt
Maths sets ppt

Sets and Functions By Saleh ElShehabey
Sets and Functions By Saleh ElShehabey

Ppt sets and set operations
Ppt sets and set operations

Set Theory
Set Theory

Elements Of Plot
Elements Of Plot

Research.report
Research.report

Descriptive Research Method
Descriptive Research Method

MCQ's for 8th class All book mxq's before mide term
MCQ's for 8th class All book mxq's before mide term

SETS [Algebra]
SETS [Algebra]

SUBJECT AND PREDICATE
SUBJECT AND PREDICATE

Subjects and predicates
Subjects and predicates

Mathematics Sets and Logic Week 1
Mathematics Sets and Logic Week 1

Maths Project on sets
Maths Project on sets

Set concepts
Set concepts

Parts Of A Book
Parts Of A Book

Mathematics class XI SETS
Mathematics class XI SETS

Elements Of Plot Cinderella
Elements Of Plot Cinderella

Plot Structure
Plot Structure

SET THEORY
SET THEORY

Descriptive Method
Descriptive Method

## Similar to 2.1 Sets

Mathematics JEE quick revision notes pdf
Mathematics JEE quick revision notes pdfgowhiksankar54

Sets functions-sequences-exercises

POWERPOINT (SETS & FUNCTIONS).pdf
POWERPOINT (SETS & FUNCTIONS).pdfMaryAnnBatac1

Set and Set operations, UITM KPPIM DUNGUN
Set and Set operations, UITM KPPIM DUNGUNbaberexha

Set
SetH K

Discrete mathematics OR Structure
Discrete mathematics OR Structure Abdullah Jan

CMSC 56 | Lecture 6: Sets & Set Operations
CMSC 56 | Lecture 6: Sets & Set Operationsallyn joy calcaben

Moazzzim Sir (25.07.23)CSE 1201, Week#3, Lecture#7.pptx
Moazzzim Sir (25.07.23)CSE 1201, Week#3, Lecture#7.pptxKhalidSyfullah6

SET AND ITS OPERATIONS
SET AND ITS OPERATIONSRohithV15

Set theory-ppt
Set theory-pptvipulAtri

JEE+Crash+course+_+Phase+I+_+Session+1+_+Sets+and++Relations+&+Functions+_+7t...
JEE+Crash+course+_+Phase+I+_+Session+1+_+Sets+and++Relations+&+Functions+_+7t...7anantsharma7

mathematical sets.pdf
mathematical sets.pdfJihudumie.Com

Final maths presentation on sets
Final maths presentation on setsRahul Avicii

### Similar to 2.1 Sets(20)

Mathematics JEE quick revision notes pdf
Mathematics JEE quick revision notes pdf

Sets functions-sequences-exercises
Sets functions-sequences-exercises

POWERPOINT (SETS & FUNCTIONS).pdf
POWERPOINT (SETS & FUNCTIONS).pdf

Set and Set operations, UITM KPPIM DUNGUN
Set and Set operations, UITM KPPIM DUNGUN

4898850.ppt
4898850.ppt

Chap2_SETS_PDF.pdf
Chap2_SETS_PDF.pdf

Sets class 11
Sets class 11

Set
Set

Discrete mathematics OR Structure
Discrete mathematics OR Structure

Mtk3013 chapter 2-3
Mtk3013 chapter 2-3

discrete maths notes.ppt
discrete maths notes.ppt

SETS-AND-SUBSETS.pptx
SETS-AND-SUBSETS.pptx

CMSC 56 | Lecture 6: Sets & Set Operations
CMSC 56 | Lecture 6: Sets & Set Operations

Moazzzim Sir (25.07.23)CSE 1201, Week#3, Lecture#7.pptx
Moazzzim Sir (25.07.23)CSE 1201, Week#3, Lecture#7.pptx

SET AND ITS OPERATIONS
SET AND ITS OPERATIONS

Set theory-ppt
Set theory-ppt

JEE+Crash+course+_+Phase+I+_+Session+1+_+Sets+and++Relations+&+Functions+_+7t...
JEE+Crash+course+_+Phase+I+_+Session+1+_+Sets+and++Relations+&+Functions+_+7t...

mathematical sets.pdf
mathematical sets.pdf

Final maths presentation on sets
Final maths presentation on sets

Set theory
Set theory

## More from showslidedump

Automated Prediction
Automated Predictionshowslidedump

2010 nasig integrating_usage_statistics
2010 nasig integrating_usage_statisticsshowslidedump

Slides Chapter10.1 10.2
Slides Chapter10.1 10.2showslidedump

5.2 Strong Induction
5.2 Strong Inductionshowslidedump

The life and_times_of_biomet_1977-2012____trine
The life and_times_of_biomet_1977-2012____trineshowslidedump

Computational Thinking
Computational Thinkingshowslidedump

### More from showslidedump(9)

Automated Prediction
Automated Prediction

2010 nasig integrating_usage_statistics
2010 nasig integrating_usage_statistics

Slides Chapter10.1 10.2
Slides Chapter10.1 10.2

5.2 Strong Induction
5.2 Strong Induction

5.1 Induction
5.1 Induction

2.2 Set Operations
2.2 Set Operations

Exam 2 Study Guide
Exam 2 Study Guide

The life and_times_of_biomet_1977-2012____trine
The life and_times_of_biomet_1977-2012____trine

Computational Thinking
Computational Thinking

ICSE English Language Class X Handwritten Notes
ICSE English Language Class X Handwritten NotesGauri S

Data Modeling - Entity Relationship Diagrams-1.pdf
Data Modeling - Entity Relationship Diagrams-1.pdfChristalin Nelson

Decision on Curriculum Change Path: Towards Standards-Based Curriculum in Ghana
Decision on Curriculum Change Path: Towards Standards-Based Curriculum in GhanaPrince Armah, PhD

spring_bee_bot_creations_erd primary.pdf
spring_bee_bot_creations_erd primary.pdfKonstantina Koutsodimou

The basics of sentences session 6pptx.pptx
The basics of sentences session 6pptx.pptxdeputymitchell2

GIÁO ÁN TIẾNG ANH GLOBAL SUCCESS LỚP 11 (CẢ NĂM) THEO CÔNG VĂN 5512 (2 CỘT) N...
GIÁO ÁN TIẾNG ANH GLOBAL SUCCESS LỚP 11 (CẢ NĂM) THEO CÔNG VĂN 5512 (2 CỘT) N...Nguyen Thanh Tu Collection

A TEXTBOOK OF INTELLECTUAL ROPERTY RIGHTS
A TEXTBOOK OF INTELLECTUAL ROPERTY RIGHTSDr.M.Geethavani

11 CI SINIF SINAQLARI - 9-2023-Aynura-Hamidova.pdf
11 CI SINIF SINAQLARI - 9-2023-Aynura-Hamidova.pdfAynouraHamidova

Understanding Canada's international higher education landscape (2024)

VPEC BROUCHER FOR ALL COURSES UPDATED FEB 2024
VPEC BROUCHER FOR ALL COURSES UPDATED FEB 2024avesmalik2

ICSE English Literature Class X Handwritten Notes
ICSE English Literature Class X Handwritten NotesGauri S

Odontogenesis and its related anomiles.pptx
Odontogenesis and its related anomiles.pptxMennat Allah Alkaram

BEZA or Bangladesh Economic Zone Authority recruitment exam question solution...
BEZA or Bangladesh Economic Zone Authority recruitment exam question solution...MohonDas

A LABORATORY MANUAL FOR ORGANIC CHEMISTRY.pdf
A LABORATORY MANUAL FOR ORGANIC CHEMISTRY.pdfDr.M.Geethavani

New Features in the Odoo 17 Sales Module
New Features in the Odoo 17 Sales ModuleCeline George

Andreas Schleicher - 20 Feb 2024 - How pop music, podcasts, and Tik Tok are i...
Andreas Schleicher - 20 Feb 2024 - How pop music, podcasts, and Tik Tok are i...EduSkills OECD

Digital Footprints to Career Pathways - Building a Strong Professional Online...
Digital Footprints to Career Pathways - Building a Strong Professional Online...Sue Beckingham

ICSE English Language Class X Handwritten Notes
ICSE English Language Class X Handwritten Notes

Data Modeling - Entity Relationship Diagrams-1.pdf
Data Modeling - Entity Relationship Diagrams-1.pdf

Decision on Curriculum Change Path: Towards Standards-Based Curriculum in Ghana
Decision on Curriculum Change Path: Towards Standards-Based Curriculum in Ghana

spring_bee_bot_creations_erd primary.pdf
spring_bee_bot_creations_erd primary.pdf

The basics of sentences session 6pptx.pptx
The basics of sentences session 6pptx.pptx

Advance Mobile Application Development class 04
Advance Mobile Application Development class 04

GIÁO ÁN TIẾNG ANH GLOBAL SUCCESS LỚP 11 (CẢ NĂM) THEO CÔNG VĂN 5512 (2 CỘT) N...
GIÁO ÁN TIẾNG ANH GLOBAL SUCCESS LỚP 11 (CẢ NĂM) THEO CÔNG VĂN 5512 (2 CỘT) N...

Lipids as Biopolymer
Lipids as Biopolymer

A TEXTBOOK OF INTELLECTUAL ROPERTY RIGHTS
A TEXTBOOK OF INTELLECTUAL ROPERTY RIGHTS

11 CI SINIF SINAQLARI - 9-2023-Aynura-Hamidova.pdf
11 CI SINIF SINAQLARI - 9-2023-Aynura-Hamidova.pdf

Understanding Canada's international higher education landscape (2024)
Understanding Canada's international higher education landscape (2024)

VPEC BROUCHER FOR ALL COURSES UPDATED FEB 2024
VPEC BROUCHER FOR ALL COURSES UPDATED FEB 2024

ICSE English Literature Class X Handwritten Notes
ICSE English Literature Class X Handwritten Notes

Odontogenesis and its related anomiles.pptx
Odontogenesis and its related anomiles.pptx

BEZA or Bangladesh Economic Zone Authority recruitment exam question solution...
BEZA or Bangladesh Economic Zone Authority recruitment exam question solution...

A LABORATORY MANUAL FOR ORGANIC CHEMISTRY.pdf
A LABORATORY MANUAL FOR ORGANIC CHEMISTRY.pdf

New Features in the Odoo 17 Sales Module
New Features in the Odoo 17 Sales Module

Andreas Schleicher - 20 Feb 2024 - How pop music, podcasts, and Tik Tok are i...
Andreas Schleicher - 20 Feb 2024 - How pop music, podcasts, and Tik Tok are i...

Digital Footprints to Career Pathways - Building a Strong Professional Online...
Digital Footprints to Career Pathways - Building a Strong Professional Online...

### 2.1 Sets

• 2. Section Summary  Definition of sets  Describing Sets  Roster Method  Set-Builder Notation  Some Important Sets in Mathematics  Empty Set and Universal Set  Subsets and Set Equality  Cardinality of Sets  Tuples  Cartesian Product
• 3. Sets  A set is collection of objects.  the students in this class  the chairs in this room  The contents have to be clearly determined  The objects in a set are called the elements, or members of the set. A set is said to contain its elements.  Set notation uses { }  The notation a ∈ A denotes that a is an element of the set A.  If a is not a member of A, write a ∉ A
• 4. Describing a Set: Roster Method  Sets are named with capital letters  Roster method is listing all elements  S = {a,b,c,d}  Order not important S = {a,b,c,d} = {b,c,a,d}  Each distinct object is either a member or not; listing more than once does not change the set. S = {a,b,c,d} = {a,b,c,b,c,d}  Elipses (…) may be used to describe a set without listing all of the members when the pattern is clear. S = {a,b,c,d, …,z }
• 5. Roster Method  Set of all vowels in the English alphabet: V = {a,e,i,o,u}  Set of all odd positive integers less than 10: O = {1,3,5,7,9}  Set of all positive integers less than 100: S = *1,2,3,…,99+  Set of all integers less than 0: S = *…., -3,-2,-1}
• 6. Tedious or Impossible  Roster method is sometime tedious or impossible  Example: (tedious)  L={x | x is a lowercase letter of the alphabet}  L= {a, b, c … z}  Example: (impossible)  Z = {x | x is a positive number}  Z = {1, 2, 3, 4 …}
• 7. Some Important Sets N = natural numbers = {0,1,2,3….+ Z = integers = *…,-3,-2,-1,0,1,2,3,…+ Z⁺ = positive integers = {1,2,3,…..+ R = set of real numbers R+ = set of positive real numbers C = set of complex numbers. Q = set of rational numbers
• 8. Set-Builder Notation  Specify the property or properties that all members must satisfy: S = {x | x is a positive integer less than 100} In word: The set S is all the elements x such that x is positive and less than 100 O = {x | x is an odd positive integer less than 10} O = {x ∈ Z⁺ | x is odd and x < 10}  Positive rational numbers: Q+ = {x ∈ R | x = p/q, for some positive integers p,q} Such thatAll Elements x
• 9. Set-Builder Notation Ex: Express Set A={x | x is a month begins with the letter M} using the roster method. Word Description Roster Method Set-Builder Notation W is the set of all days of the week W= {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday} W={x | x is a day of the week}
• 10. Some things to remember  Sets can be elements of sets. {{1,2,3},a, {b,c}} {N,Z,Q,R}  The empty set is different from a set containing the empty set. ∅ ≠ * ∅ + { }
• 11.  Example: Recognize the empty set: 1. {0} No 2. 0 No, it is not a set 3. {x | x is a number less than 4 and greater than 10} No 4. {x | x is a square with 3 sides} No
• 12. Notations for Set Membersship  ∈  Is an element of  ∈  Is not an element of  Example: Determine if each statement is true or false.  r ∈ *a, b, c … z+  True  7 ∈ *1, 2, 3, 4, 5+  False  *a+ ∈ *a, b+  False
• 13. Set Equality Definition: Two sets are equal if and only if they have the same elements.  Therefore if A and B are sets, then A and B are equal if and only if .  We write A = B if A and B are equal sets. {1,3,5} = {3, 5, 1} {1,5,5,5,3,3,1} = {1,3,5}
• 14. Subsets Definition: The set A is a subset of B, if and only if every element of A is also an element of B.  The notation A ⊆ B is used to indicate that A is a subset of the set B.  A ⊆ B holds if and only if is true.
• 15. Showing a Set is or is not a Subset of Another Set  Showing that A is a Subset of B: To show that A ⊆ B, show that if x belongs to A, then x also belongs to B.  Showing that A is not a Subset of B: To show that A is not a subset of B, A ⊈ B, find an element x ∈ A with x ∉ B. (Such an x is a counterexample to the claim that x ∈ A implies x ∈ B.) Examples: The set of all computer science majors at your school is a subset of all students at your school.
• 16. Another look at Equality of Sets  Recall that two sets A and B are equal, denoted by A = B, iff  Using logical equivalences we have that A = B iff  This is equivalent to A ⊆ B and B ⊆ A
• 17. Proper Subsets Definition: If A ⊆ B, but A ≠B, then we say A is a proper subset of B, denoted by A ⊂ B. If A ⊂ B, then is true.
• 18. Set Cardinality Definition: If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is finite. Otherwise it is infinite. Definition: The cardinality of a finite set A, denoted by |A|, is the number of elements of A. Examples: 1. |ø| = 0 2. Let S be the letters of the English alphabet Then |S| = 26 3. C= {1,2,3} |C|=3 4. D= {1, 1, 2, 3, 3} |D|= 3 5. E={ø} |E|= 1 The set of integers is infinite. Finite: ends Infinite: no end
• 19. Universal Set and Venn Diagram  The universal set U: is the set containing everything currently under consideration.  Contents depend on the context.  Venn Diagram:  visual relationship among sets  Uses circles and rectangles U Venn Diagram a e i o u V John Venn (1834-1923) Cambridge, UK
• 20. Example Use the Venn Diagram to determine each of the following sets: a) U U={\$, 5, s, w} b) B B={3} U B 5 \$ 3 s w
• 21. Representing 2 sets in a Venn Diagram  4 different ways U A Disjoint Set Proper Subset U AB B U A=B Equal Sets Set with some common elementsU A B
• 22. Example Use the Venn Diagram to determine each of the following sets: a) U U={a, b, c, d, e, f} b) B B={c, d} c) The set of elements in A but not B {a, b} d) The set of elements in U but not B {a, b, e, f} e) The set of elements both in A and B {c} U B c b a f e A d
• 23. Example Use the Venn Diagram to illustrate the relationship: A ⊆ B and B ⊆ C U BA C
• 24. Power Sets Definition: The set of all subsets of a set A, denoted P(A), is called the power set of A. Example: If A = {a,b} then P(A) = {ø, {a},{b},{a,b}}  If a set has n elements, then the cardinality of the power set is 2ⁿ.
• 25. Cartesian Product Definition: The Cartesian Product of two sets A and B, denoted by A × B is the set of ordered pairs (a,b) where a ∈ A and b ∈ B . Example: A = {a,b} B = {1,2,3} A × B = {(a,1),(a,2),(a,3), (b,1),(b,2),(b,3)}  Definition: A subset R of the Cartesian product A × B is called a relation from the set A to the set B. René Descartes (1596-1650)
• 26. Cartesian Product Example: What is A × B × C where A = {0,1}, B = {1,2} and C = {0,1,2} Solution: A × B × C = {(0,1,0), (0,1,1), (0,1,2),(0,2,0), (0,2,1), (0,2,2),(1,1,0), (1,1,1), (1,1,2), (1,2,0), (1,2,1), (1,1,2)}
• 27. Let A ={a, b, c}, B= {x, y}, and C={0, 1} Find: a) A × B × C {(a, x, 0), (a, x, 1), (a, y, 0), (a, y, 1), (b, x, 0), (b, x, 1), (b, y, 0), (b, y, 1), (c, x, 0), (c, x, 1), (c, y, 0), (c, y, 1)} b) C × B × A {(0, x, a), (0, x, b), (0, x, c), (0, y, a), (0, y, b), (0, y, c), (1, x, a), (1, x, b), (1, x, c), (1, y, a), (1, y, b), (1, y, c)}
Current LanguageEnglish
Español
Portugues
Français
Deutsche