Venn-diagrams

Actually Venn is a mathematician surname called Euler john Venn who first introduced a diagram to represent the sets and its subset. So that diagram is called Venn diagram. So Venn diagram is a diagram like a rectangle, circle or oval shapes used for representing sets. Generally Rectangle represent Universal set and Circles or Ovals inside the rectangle shows its subset.

Following are some Venn diagram which represent subset, disjoint and overlapping set       

i)  AÌU (A is a subset of U).

ii)     A and B are disjoint sets. 

         






                             

iii)    A and B are overlapping sets.     

  











Some Information about John Venn

                                                                               

v  A British logician and philosopher                                  

v  Born : 1834 August 4, Hull, London

v  Famous for Venn diagram

v  Degree in Mathematics (1857) at 23 years old.

v  Notable awards : Fellow of the royal society

v  Died: 1923 April 4 at 88 years old

                                                                                                 

Set operation

There are four main set operations. They are:

Ø  Union of two or more sets[È]

Ø  Intersection of two or more sets[Ç]

Ø  Difference of two sets[-]

Ø  Complement of a set[`A, A^c or A¢]

1)      Union of two or more sets

It means the set of all members which belongs to either single set or to all set.

In two set, Let A and B

The member of the union of these two set should belong to either A or B or both and denoted by AÈB.

In three set, Let A, B and C

The member of the Union of these three sets should belong to either A or B or C or All of three and denoted by AÈBÈC.

In this way we can define for n sets. But in exam we write only for two sets.

There will be two conditions

i) When they are overlapping the repeated member should be written only one time

Consider overlapping set A and B where A= {a,b,c} and B={c,d,e}

Then the union of these two sets A and B symbolically AÈB is {a,b,c,d,e}. here c is repeated so c is taken only one time not repeatedly.

ii) When two are disjoint the problem will not arise.

Consider disjoint set A and B where A={a,c,d} and B={b,e,f}

Then the union of these two sets A and B symbolically AÈB is {a,b,c,d,e,f}

This is an example of two sets. Similarly we can make union of more than two sets when they are overlapping or disjoints.

Generalized Notation : AÈB = {x:xÎA or xÎB}

Representation in Venn diagram when they are overlapping and disjoint.

v  When set A and B are disjoint, then AÈB can be represented in this way by shaded region

v  When set A and B are overlapping, then AÈB can be represented in this way by shaded region.

v  When three sets are given i.e. A, B and C and they are disjoint, then AÈBÈC can be represented         in this way by shaded region

v  When three sets are given i.e A, B and C and they are overlapping, then AÈBÈC can be                       represented in this way by shaded region

Intersection of two or more sets:

It means the set of all members common to all sets.

For two sets

So the intersection of two setA and B  is the set of all members that common to both A and B. It is denoted by AÇB

For three sets

The intersection of three sets A,B and C is the Set of all members that common to all A,B and C. It is denoted by AÇBÇC.

In this way we can define for n sets. But in exam we write only for two sets.

Well in the intersection case, the set is always empty in disjoint set. so it is always exercised in overlapping set.

Representation by Venn diagram through shaded region.

v  When two sets are intersecting.

                                                              

   v  When two sets are disjoint.

                                                                    U

Since the element common to them is zero so there is no shaded region for AÇB.

v  When three sets are overlapping.

                                                                                          

                                             

Difference of two sets

The difference of two set A and B, denoted by A-B is the set of the elements of A which do not belong to B.

For example if A= {1,2,3,4,5}, B={1,3,5,7,9} then A-B={2,4} since 1,3,5 are also in B so it is not collected and the element 7, 9 is ignored, so the remaining element 2,4 are the elements of the set A-B. Similarly we get B-A= {7,9}

Generalized Notation: A-B= {x: xÎA, but xÏA} similarly B-A= {x:xÎB, but xÏA}

Representation on Venn diagram

1.      
       Disjoint case

                                                                            A-B                                                                  B-A

2.      Overlapping case

                     A-B                                                               B-A

Complement of a set

If A be a subset of a universal set U, then the compliment of A, denoted by`A or A^c or A¢ is set of the elements of U which do not belong to P.

It is the difference of a two sets where a set from which another set is deducted  is universal set and the another set is its subset.

        i.e.             

`A or Ac or A¢ = U-A

For example:

If U= {1, 2, 3, 4……………….10} and A= {1,4,5,7}

Then`A= U-P

              = {1,2,3,4………….10} – {1,4,5,7}

              = {2,3,6,8,9,10}

Generalized Notation : `A ={x:xÎU, but xÏA}

Representation on Venn diagram

                                       

                  

 

                                 

Practice Test on Operations on Sets

1. If A = {2, 3, 4, 5}     B = {4, 5, 6, 7}     C = {6, 7, 8, 9}     D = {8, 9, 10, 11}, find 

(a) A ∪ B 

(b) A ∪ C 

(c) B ∪ C 

(d) B ∪ D 

(e) (A ∪ B) ∪ C

(f) A ∪ (B ∪ C) 

(g) B ∪ (C ∪ D) 


2. If A = {4, 6, 8, 10, 12} B = {8, 10, 12, 14} C = {12, 14, 16} D = {16, 18}, find

(a) A ∩ B

(b) B ∩ C

(c) A ∩ (C ∩ D) 

(d) A ∩ C

(e) B ∩ D

(f)(A ∩ B) ∪ C 

(g) A ∩ (B ∪ D) 

(h) (A ∩ B) ∪ (B ∩ C) 

(i) (A ∪ D) ∩ (B ∪ C) 


3. If A = {4, 7, 10, 13, 16, 19, 22}   B = {5, 9, 13, 17, 20}
C = {3, 5, 7, 9, 11, 13, 15, 17}   D = {6, 11, 16, 21} then find 

(a) A - C 

(b) D - A

(c) D - B

(d) A - D 

(e) B - C

(f) C - D 

(g) B - A 

(h) B - D 

(i) D - C 

(j) A - B

(k) C - B

(l) C - A

More Practice Test on Operations on Sets
4. If A and B are two sets such that A ⊂ B, then what is A∪B? 

5. Find the union, intersection and the difference (A - B) of the following pairs of sets. 

(a) A = The set of all letters of the word FEAST 

     B = The set of all letters of the word TASTE 

(b) A = {x : x ∈ W, 0 < x ≤ 7} 

     B = {x : x ∈ W, 4 < x < 9} 

(c) A = {x | x ∈ N, x is a factor of 12} 

     B = {x | x ∈ N, x is a multiple of 2, x < 12} 

(d) A = The set of all even numbers less than 12 

     B = The set of all odd numbers less than 11 

(e) A = {x : x ∈ I, -2 < x < 2} 

     B = {x : x ∈ I, -1 < x < 4} 

(f) A = {a, l, m, n, p} 

    B = {q, r, l, a, s, n} 


6. Let X = {2, 4, 5, 6}   Y = {3, 4, 7, 8}   Z = {5, 6, 7, 8}, find

(a) (X - Y) ∪ (Y - X) 

(b) (X - Y) ∩ (Y - X) 

(c) (Y - Z) ∪ (Z - Y) 

(d) (Y - Z) ∩ (Z - Y) 

Practice Test on Operations on Sets

7. Let ξ = {1, 2, 3, 4, 5, 6, 7} and A = {1, 2, 3, 4, 5} B = {2, 5, 7} show that 

(a) (A ∪ B)' = A' ∩ B'

(b) (A ∩ B)' = A' ∪ B'

(c) (A ∩ B) = B ∩ A 

(d) (A ∪ B) = B ∪ A

8. Let P = {a, b, c, d}   Q = {b, d, f}   R = {a, c, e} verify that 

(a) (P ∪ Q) ∪ R = P ∪ (Q ∪ R) 

(b) (P ∩ Q) ∩ R = P ∩ (Q ∩ R) 


Answers for practice test on operations on sets are given below to check the correct answers. 

Answers: 

1. (a) {2, 3, 4, 5, 6, 7} 
(b) {2, 3, 4, 5, 6, 7, 8, 9} 
(c) {4, 5, 6, 7, 8, 9} 
(d) {4, 5, 6, 7, 8, 9, 10, 11}
(e) {2, 3, 4, 5, 6, 7, 8, 9} 
(f) {2, 3, 4, 5, 6, 7, 8, 9} 
(g) {4, 5, 6, 7, 8, 9, 10, 11} 

2. (a) {8, 10, 12}
(b) {12, 14}
(c) ∅
(d) {12} 
(e) d 
(f) {8, 10, 12, 14, 16} 
(g) {8} 
(h) {8, 10, 12, 14} 
(i) {8, 10, 12, 16} 


3. (a) {4, 10, 16, 19, 22} 
(b) {6, 11, 21} 
(c) {6, 11, 16, 21}
(d) {4, 7, 10, 13, 19, 22}
(e) {20} 
(f) {3, 5, 7, 9, 13, 15, 17} 
(g) {5, 19, 17, 20} 
(h) {5, 9, 13, 17, 20} 
(i) {6, 16, 21}
(j) {4, 7, 10, 16, 19, 22} 
(k) {3, 7, 11, 15} 
(l) {3, 5, 9 11, 15, 17}


4. B 

5. (a) {F, E, A, S, T}, {E, A, S, T}, {F} 
(b) {1, 2, 3, 4, 5, 6, 7, 8}, {5, 6, 7}, {1, 2, 3, 4} 
(c) {1, 2, 3, 4, 6, 8, 10, 12}, {2, 4, 6}, {1, 3, 12} 
(d) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, d, {2, 4, 6, 8, 10} 
(e) {-1, 0, 1, 2, 3}, {0, 1}, {-1}
(f) {a, 1, m, n, p, q, r, s}, {a, l, n}, {m, p} 


6. (a) {2, 3, 5, 6, 7, 8} 
(b) d 
(c) d {3, 4, 5, 6}
(d) d


7. (a) L.H.S. = R. H. S = {6} 
(b) L.H.S. = R. H. S = {1, 3, 4, 6, 7} 
(c) {2, 5} 
(d) {1, 2, 3, 4, 5, 7}


8. (a) {a, b, c, d, e, f} 
(b) d

                   

Introdution to SET

Before studying this subject let’s read about the numbers and its type

# Natural number : the counting number that we use in general is called natural number. i.e. 1, 2,3,4,5,…………….. Generally denoted by N.

# Whole number: the natural number including 0 is the whole number.

 i.e. 0,1,2,3,4,…………………… generally denoted by W.

# Positive number: the number whose sign and value is positive i.e +1,+2,+3,……..

Generally in positive number we don’t write its sign so 1,2,3……. Itself represents positive number.

# Negative number : the number whose sign and value is negative i.e.    -1,-2,-3,……………..

# Integer : the set of whole number and negative number is called integer. i.e …….-3,-2,-1,0,1,2,3………. Generally denoted by I no Z.

# Even number : the number which is divisible by 2 is called even number i.e 2,4,6,8………….. generally denoted by E.

# Odd number : the number which is not divisible by 2 is called odd number. i.e. 1,3,5,7,9,……………… generally denoted by O.

# Prime number: the number whose factor is only 1 is called prime number. i.e. 2, 3,5,7,11,………………..  Generally denoted by P

Here 2 have only one factor 1 i.e 2x1, similarly 5x1,11x1…………..and so on.

Set review

Set→ collection of well defined Object

→ it means the object which we collect must be well defined to be a set.

→so the next question is “what is well-defined object ? ”

→ the well defined object means that object which could be easily identifying and easily distinguishable

             Identifying means which is easily known by us

             Distinguishable means we can differentiate it or apart them by its distinct/special 

             Properties

For example,

       The collection of natural numbers

          i.e. 1, 2, 3, 4, 5 …

here 1, 2, 3, 4, 5 , these numbers are easily known so there is no difficulty to identify it and these numbers have different value so they can be parted hence it is a set

Similarly take another example

        The collection of vowel sounds

         i.e. a, e, i, o, u

Now they are the alphabets of English language system so it is already known for all of you.

And does ‘a’ give the same meaning of e or e to i  or  i to o and so on…

No ….. it has different meaning and different pronunciation 

    Can we write apple to eppla ?

                 No so these alphabets have distinct characteristics.

So it is also a set.

*******

Now the big question is “Which type of collection is not a set?”

The simple answer is those set in which object cannot be identifying or distinguished that means non well defined object

For example

Ram is taller than shyam

Mohan is taller than ram

If we gonna collect the tall boy with reference to above given information, then I wanna ask one question here

   Actually who is the tall student?

Is ram or mohan or both?

When we collect ram as a tall student but as compared to mohan it will be contradiction (wrong) why because ram is not taller than mohan but he is shorter than him.

So our collection becomes wrong.

Similarly take another example

The collection of sweet fruits

We all know that apple, orange, grapes etc are the sweet fruits if they are already ripe.

Here how can we say that apple is sweeter than orange or orange is sweeter than grapes?

Because it differ or depend upon the taste of different people so we cannot identify it and also differentiate it so it is not a set.

But if the dimension of tallness of people is given then it will be a set

Now the collection of tall student more than height of 2m, 3m and so on is Set.

Set notation

Now let’s go to set notation

Simply it is about presenting means how we present the set and the object which we collect.

Sets are denoted by capital letter i.e. A,B,C,D,E………………….. and the object which we collect is said to be element or member of that set and it is denoted by small letters. And the element or members is denoted by small letter.

For example

The collection of vowel sound

                   A= {a,e,i,o,u}

Next lesson is Î and Ï

here Î                              Ï

          ¯                              ¯

      Belongs to                       Doesnot belong to

For example 2ÎN

                Read as 2 belongs to N

Which means 2 is a member of N

                      Or 2 is an element of N

Again   3ÏE

              Read as 3 doesn’t belong to even number i.e. E

Which means 3 is not a member of E

                        Or 3 is not an element of E.

Specification of sets

                  Or

Description of sets

Generally we describe set in three ways

1)      Description method Þ  described by words

e.g. A set of odd number less than 10

        A set of 8 planet of solar system

2)      Listing or roster method Þ described by listing elements inside braces { }.

          A= {1, 3, 5, 7, 9}

          B= {Mercury, Venus, Mars, Jupiter, Earth, Saturn, Uranus, Neptune}

One general question I want to ask all of you that “which is the planet taken away from the nine planets?”

Yes it is Pluto.

3)      Set-builder or Rule method [:]Þ described by representing elements by a variable mentioning their common properties.

For example A={x: xÎodd number<10}

                       B= {x:x is a planet of solar system}

Well, we use also this method on set operation for generalized notation. Mainly other two methods are less used than it.

Types of set [on the basis of elements]

1)      Null or empty set

Þ Having no element

Þ Denoted by f (phi) or by empty braces { }

Þ e.g. the set of natural number between 1 & 2

2)      Unit or singleton set

Þ The set having only one element

Þ e.g. the set of natural number between 2 & 4

      or N= {5},     A= {e}

3)      Finite set

ÞHaving countable number of elements

Þe.g. the set of English alphabet

             the set of points in a circle why because it is a bounded figure

4)      Infinite set

Þ having infinite number of elements

Þ e.g. the set of natural number

              i.e. 1,2,3……….999,……..12345,………

              the set of points in a arbitrary straight line why because it is unbounded figure.

Cardinal number of sets

The number of elements in a set is called its cardinal number and it is denoted as n (A).

For example

                         If A= {a, e, i, o, u} then n (A)=  5

Set relations

Equal(=)	Equivalent(~)	Overlapping	Disjoints
·         Two or more than two sets have exactly the same element but the order may differ For example ·         A= {3,6,9,12} ·         B= {6,3,12,9} ·         Then A= B	·         Two or more than two sets have the equal cardinal number. For example ·         A ={3, 6, 7, 9} ·         B = {a, e, i, o} ·         Then A~B	·         Two or more than two sets have at least one element ·         For example ·         A={1,2,3,4,5,6} ·         B= {2,4,7,8,9} ·         In above 2 and 4 are both in A and B, so A &B are overlapping set. ·         The set of people and the set of boys are the another example.	·         Two or more than two sets have no common element ·         For example ·         A={1,2,3} ·         B={4,5,6} ·         In above none of the element are same. So it is. ·         The set of boys and the set of girls.

Sub sets[ Í ]

In two set A and set B, set B is said to be a subset of A if every element of set B is contained by the set A. It means the object which we collect in set B must be the element of set A to be subset. And we denote it by Í. Remember that f is a subset of every set.

For example

A ={1,2,3,4,5}                   C={a,b,c,d,e}

And B={2,3,4}                   D={b,c,e,a,d}

Then BÍA                          Then DÍC

Proper subset[Ì]

Considering above two set A and B if B is a subset of A and B¹A, then B is said to be proper subset of A. It means the subset B has at least one element less than the set A. It is denoted by BÌA

For example

A={1,2,3,4,5}       

B={3,4,1,2}

C={1,5,2,}

BÌA because B lack one element of A i.e. 5

CÌA because C lack two element of A i.e. 3,4

Let D = {2, 3, 4, 7}

Is DÌA or DÍA?

The answer is no because the set D does not contain all element of A i.e 7 so it is neither subset nor proper subset.

Universal set [U]

A set under the consideration from which many subsets can be formed is called a universal set.

Let U be a set of natural number

 i.e U= {1,2,3,4,…………………………}

Considerations                                   Subsets

Even number(E)                                {2,4,6,…………….}

Odd number (O)                               {1,3,7,…………….}

Multiple of 3 (M)                              {3,6,9,……………..}

………………..so on

So E, O,M,….. are the subsets under the consideration i.e. even number, odd number, multiple of 3 and so on and these all sets are subsets of U. Hence U is a universal set.

*******

Difference between Ì and Í

Simply Í is a mixture of Ì and =.

So we can say that Í can be written as Ì

                            But Ì cannot be written as Í

And remember Í symbolize subset and Ì symbolize only proper subset.

Let us give an example

A= {1,2,3,4,5}

B= {4,2,1,3,5}

C= {4,2,1}

D= {1,3,5}

E= {2,4,5,1,3}

Results.

      A is subset of B but A is not a proper subset of B because A=B.

      B is subset of A but B is not a proper subset of A because B=A.

      A is neither a subset of C nor a proper subset of C because A is universal for C

      But C is both subset and proper subset of A because C¹A

      Same case in D

      It is same as B.

Exercise 1.1

1.      If A={1,2,3,4,5}, whether 3,5 and 8 are the members of the set A or not, write with set notation symbols.

2.   A set of odd numbers between 0 and 10. Express it in listing method  and set builder method.

3.      If Z= {x:x is an integer, -1£x£1}, express it in description method and listing method.

4.      State whether the following sets are null, unit, finite or infinite sets

a)      The set of the highest peak of mountain .

b)      The set of whole numbers more than 10.

c)      The set of whole numbers between 10 and 100.
d)      The set of prime numbers between 14 and 16.

** Prime numbers are that numbers whose factor is only 1.

5.      If A = {x:x+1£5,xÎW}, find n (A)

6.      Whether the sets P = {3,6,9,12} and Q= {1,3,5,7} are equal or equivalent ? Write with reason.

7.  State whether the sets X = {2,4,6,8} and Y= {6,4,2,8} are equal or equivalent ? Give reason.

8.      State whether the sets A= {a,b,c,d,e} and B= {a,e,i,o,u} are  overlapping or disjoint ? Write with reason.

9.      State whether the sets Z= {-2,-1,0,1,2} and N= {3,4,5,6} are overlapping or disjoint ?

10. If U= {1,2,3,………15}, A= {1,3,5,7,9}, B= {1,2,3,……15} and C= {4,8,12,16}, answer the following questions.

a.      Is A the subset of U? Give reason.

b.      Is B the proper subset of U? Give reason.

c.       Is CÍU? Give reason.

11. Write the possible subsets of the set A = {3,6,9}

12. What can be the universal sets from which the following subsets can be formed ?

a.      The set of cricket team of a school.

b.      The set of literate women of a village.

c.       The set of even numbers.

See the solution for this exercise