Q. What is Set?
A. A set is a defined section of objects. These objects are called elements of the set.
Q. How number of elements in a Set are denoted?
A. Number of elements in Set A is denoted by n(A) and are called cardinal numbers.
Q. How set is denoted?
A. Set is denoted by a capital letter and elements of a set is denoted by small letters.
Q. How set is represented?
A. A set can be represented in two ways.
- Roster form: By listing all it's elements enclosed in {curly brackets} and commas{x,y} are used to separate each element.
EX: A = set of odd numbers lesser than or equal to 21 can be represented as
A = {1,3,5,7,9,11,13,15,17,19,21}
- Set builder form: By defining the specific property that determines the elements of set.
EX: B = set of numbers divisible by two and lesser than 20 can be represented as
B = {a : a is an even number lesser than 20}
Q. What are the type of Sets?
A. Sets are generally defined on the basis of their elements.
- Finite Set: A set consisting a definite number of elements. Ex: A is set of vowels in English language. A = {a.e.i.o.u}
- Infinite Set: A set which is not finite. Ex. B is set of arcs that can be drawn in a circle.
- Empty Set: A set that has no element at all. Ex: C is a set even prime numbers greater that 11. Then C={}
- Singleton Set: A set containing only one element. Ex: D = even prime numbers. Then D = {2}.
- Equal Set: When two sets have exactly same elements. Ex:
Set A has all the even numbers less than 20. => A = {2,4,6,8,10,12,14,16,18}
Set B has all the number divisible by 2 and lesser than 20. => B = {2,4,6,8,10,12,14,16,18}
All the elements of set A and B are same, hence A and B are equal.
- Sub-set & Super-set: When all the elements of Set 'A' are the elements of Set 'B', A will be called the subset of B. B will be called Superset of A. Ex:
Set A has all the even prime numbers. => A = {2}
Set B has all the even numbers less than 10.=> A = {2,4,6,8}
All the elements of set A are the elements of B but B has some elements that are not there in A. Hence A is a sub-set of B and B is a superset of A.
- Universal set: A set that is the super set for more than 1 set.Ex:
Set A has all Prime numbers => A = {2,3,5,7,11,13......}
Set B has all even numbers => B = {2,4,6,8,10,12,14...}
Set C has all odd numbers => C = {1,3,5,7,9,11...........}
Set U has all integers => U = {1,2,3,4,5,6,7,8,9,10........}
Set U has all the elements that SET A B C may have. Hence Set U is the universal set for Sets A, B, C.
- Complementary set: Set having all remaining elements which a particular set is not having in comparison to it's universal set and represented with A'.
Set U has all the integers. => U = {1,2,3,4,5,6,7,8,9,10......}
Set A has all the even numbers. => A = {2,4,6,8,10............}
Set A' has all the odd numbers. => A' = {1,3,5,7,9,11..........}
Set A' has all the elements that were missing in the Set A. Hence Set A' is complement of Set A.
Q. How to solve the questions that are formed from 'Set theory'?
A. The questions from the set theory can be solved
- Mostly with the help of Venn diagrams.
- With the help of Set Theory Laws.
Q. How to represent a set with the help of Venn diagram?
A. Venn diagram uses simple geometric shapes to represent different sets
- B subset of A can be represented as
Where U = universal set
- Complement of Set A can be represented as
Where U = universal set.
Q. What are the Laws in Set Theory?
A. There are two basic operations on which Set Laws are based
- Union of Set: A set that has all the elements of two or more sets. Union is denoted with 'U'
A U B = {elements of A, elements of B}
Ex. A={1,2,3,4,5} and B = {3,4,5,6,7}
A U B = {1,2,3,4,5,6,7}
- Intersection of Set: A set that has common elements of two or more sets. Intersection is represented as 'Ω' (inverse of U)
A Ω B = {common elements of Set A and Set B}
Ex. A = {1,2,3,4,5} and B = {3,4,5,6,7}
A Ω B = {3,4,5}
If A, B and C are three sets then <!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><!--[if !mso]><object classid="clsid:38481807-CA0E-42D2-BF39-B33AF135CC4D" id=ieooui></object> <style> st1\:*{behavior:url(#ieooui) } </style> <![endif]--><!--[endif]-->
<!--[if !supportLists]-->1. <!--[endif]-->1. A U (B Ω C) = (A U B) Ω (A U C)
To understand the concept, let's see the same with the help of a Venn diagram.
<!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:DoNotShowRevisions/> <w:DoNotPrintRevisions/> <w:DoNotShowMarkup/> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman";} </style> <![endif]-->
<!--[if !supportLists]-->2. <!--[endif]-->2. A Ω (B U C) = (A Ω B) U (A Ω C)
To understand the concept, let's see the same with the help of a Venn diagram.
<!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:DoNotShowRevisions/> <w:DoNotPrintRevisions/> <w:DoNotShowMarkup/> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman";} </style> <![endif]-->
3. Difference of
Sets:
If A and B are two sets, then all the elements that belong to Set A but do not belong to Set B is called the difference (A - B).
<!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:DoNotShowRevisions/> <w:DoNotPrintRevisions/> <w:DoNotShowMarkup/> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:Calibri; mso-fareast-font-family:Calibri;} </style> <![endif]-->
Venn diagram uses simple geometric shapes to represent different sets.
De-Morgan's Laws: There are two identities involving Union, Intersection and Complement.
- <!--[endif]-->(A U B)' = A' Ω B'
- (A Ω B)' = A' U B'
Examples Set Theory
Let's practice some questions based on Set theory.
- Two finite Sets A and B, have x and y number of elements. The total number of subsets of Set A is 16 times the number of Set B.<!--[if !supportLists]--> Find the value of x-y.
(Solution)
Number of subsets of the Set A = 2x
Number of subsets of the Set A = 2y
As per question ( 2x / 2y) = 16
2x-y = 16
2x-y
=
24
x-y = 4
- <!--[if !supportLists]-->If A = {x | 3x2 - 7x - 6 =0} and B = { x | 6x2 - 5x - 6 =0}
(Solution)
3x2
-
7x
-
6
=0
(3x
-
2)(x
-
3)
=
0
X = -2/3 ; 3
A = {-2/3 , 3}
6x2 - 5x - 6 =0
(3x + 2)(2x - 3) = 0
X = -2/3 ; 3/2
B = {-2/3 , 3/2}
Therefore, A Ω B = {-2/3}
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21 Comments
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gauravjain26 said – Thu, 18 Sep 2008 18:44:54 -0000 ( Link )
Yes this is! and if you will give some time to read other lessons on quant section, i am sure you will end up leaving the same comment.
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panthou-luwang said – Mon, 15 Dec 2008 07:12:45 -0000 ( Link )
good!! next time lets go for tougher problems..
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kartheekgoli said – Wed, 07 Jan 2009 12:31:33 -0000 ( Link )
It’s Very fyn..! 4m the basics..! But Can u provide some more examples..??
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Sureshbala said – Fri, 09 Jan 2009 06:26:26 -0000 ( Link )
Dear Kartheek,
Please go through other lessons as well and very soon we are going to publish advanced concepts from Set Theory
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rammystery said – Mon, 12 Jan 2009 03:18:41 -0000 ( Link )
Thanks a lot.. Great effort… U ve started it from the basics.. great job.. But it will be better if u provide more questions on this..
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ankit20188 said – Sat, 17 Jan 2009 06:50:14 -0000 ( Link )
Thanks a lot.. learnhub .i m very impresive your fecilities .i peactice everyday a new lesson
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asureshwaran said – Wed, 04 Feb 2009 02:57:19 -0000 ( Link )
well. this lesson is good. i think you can improve it further.
include more questions on graphs and data interpretation questions man. ETS asks such irritating questions on these topics. if time is sufficient, we can crack it well. if not we end up with quant scores of 710 or 730 only.
asureshwaran
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Balaji Hariharan said – Sun, 08 Feb 2009 16:21:48 -0000 ( Link )
Thiz link is good! but is tat posbl for to gv any link related to Probability with sets…..
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deepakkumar753 said – Thu, 19 Mar 2009 10:31:38 -0000 ( Link )
really a good approch to clearing the topic.thanx to author
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amitrajlaxmi said – Fri, 14 Aug 2009 08:37:38 -0000 ( Link )
easy to understand…..........but add more examples
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amitrajlaxmi said – Sun, 23 Aug 2009 09:37:56 -0000 ( Link )
was really helpful…......but give more examples
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POOJACHAUDHARY said – Fri, 18 Sep 2009 09:53:16 -0000 ( Link )
thanxxx a lot sir its vry benificial lesson POOJA CHAUDHARY
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vishnus1988 said – Sun, 20 Sep 2009 16:13:29 -0000 ( Link )
some of the portions are worst readable
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