Thursday, 4 May 2017

How to Solve Set Theory Questions using Set Theory Formulas For IBPS PO Exam

In this post, we will solve Set Theory Questions using Set Theory Formulas for IBPS PO Exam
Set Theory Formulas


Set Theory Questions are generally asked for one mark or five marks in  IBPS PO Exam. Set Theory concept is a very simple concept which can easily fetch you marks within two minutes. Set Theory Questions can simply be solved by just substituting the values in the Set Theory Formulas.

In this post will discuss What is Set Theory? What are different types of Sets? What are the elements of sets? How to derive Set Theory Formulas and How to Solve Set theory Problems?

What is Set Theory?


Set theory is a branch of mathematical logic which is used to study sets. A set is well defined collections of objects.

What are Infinite Sets?


Infinite Sets are those sets which have infinite objects or infinite elements.

Example of Infinite Sets


=> Set of natural numbers (N) = [ 1, 2, 3, 4, 5... ]
=> Set of whole numbers (W) - [ 0,1, 2, 3, 4....]
=> Set of Integers (I) = [ ...-2, -1, 0, 1, 2, 3....]
 => Set of odd numbers =  [1, 3, 5, 7, 9]
=> Set of even numbers = [ 2, 4, 6, 8,]

What are Finite Sets?


Finite Sets are those sets which have finite elements or finite objects.

Example of Finite Sets


=> Group A = [ 1, 2, 3, 4, 5 ] 
In group A there are only 5 elements - 1, 2, 3, 4 and 5
=> Group B = [ 2, 3, 4, 5, 7, 8 ]
In group B there are 6 elements - 2, 3, 4, 5, 7 and 8

What are the Operations that can be performed on two different sets?


Sets: A = { 1, 2, 3, 4, 5} ; B = { 3, 4, 5, 7, 9 } ; C = { 9, 12, 15, 18, 21}

1. Union of sets: Union of two sets A and B, denoted by AUB is the set of all elements either in A or in B or in both A and B. Therefore, AUB = 1,2,3,4,5,7,9
It is the type of set in which all the elements from both the sets are written.

2.Intersection of sets: Intersection of two sets A and B, denoted by AnB is the set of all elements that belong to both A and B. Therefore,  AnB = 3, 4, 5
In this type of set all the elements that are common in A and B are written.

3.Disjoint Sets: Two sets A and B are said to be disjoined if AnB = Ø.
In the above given sets A and C are disjoint sets as they share no common element in them. Disjoint sets are also called as Null sets.
AnC = Ø , hence A and C are disjoint , if AnB = Ø

4.Difference of sets: Difference of two sets A and B , denoted by (A - B) is set of elements that belong to A but not B.
This means that ( A - B ) is not equal to (B - A)
For Example: ( A - B ) = { 1, 2, 3, 4, 5} - { 3, 4, 5, 7, 9 }  = { 1, 2 }
(B - A) = { 3, 4, 5, 7, 9 } - { 1, 2, 3, 4, 5} = { 7, 9}

What is a Venn Diagram?

Operation of sets can be represented as diagrams. A universal set is represented by rectangle and a sub-set is represented by a circle in the rectangle. A venn diagram shows all the possible logical relationship between a finite collection of sets.
Set Theory Formulas
Set Theory Formulas

Set Theory Formulas
Total = a + b + c + d
Hockey = a + b ; Only Hockey = a
Cricket = b + c ; Only Cricket = c
Both Hockey and Cricket = b
Neither Hockey nor Cricket = d
Set Theory Formulas
Example 1:
Question: In a class of 120 students , 70 students passed in English , 80 students passed in Hindi and 40 students passes in both English and Hindi. How many students failed in both the students?

Solution:
Step 1: 
To solve Set theory Questions by Using Set theory formulas, we need to first draw a venn diagram. 
First draw a rectangle which represents total number of students, then draw two circles which intersects each other. The region which represent number of students who passed in English label it as 'a', the region which represents number of students who passed in Hindi label it as 'c' and the intersection region which represents total number of students who passed in both English and Hindi label it as 'b.'
Set Theory Formulas


Step 2:
By substituting the value sin the set theory formulas,we get
Total = A + B - Both AB + none (AB)
120 = 70 + 80 - 40 + none
None = 120 - 110
None = 10 students

Therefore, there are 10 students who neither passed in English nor Hindi.

How to solve Set Theory Questions using Set Theory Formula when Three Sets are given.

In this module we will discuss, how to draw venn diagrams when three sets are given and solve this Set Theory Questions by using Set theory formulas.

In a venn diagram, A rectangle represents the universal sets and the circles represent the sub sets, the overlapping of the circles shows the intersection of two sets.

Set Theory Formulas

A = a + d+ e + g ; Only A = a
B = b + d + f + g ; Only B = b
C = c + e + f + g ; Only C = c
Both (AB) = d + g
Both (AC) = e + g
Both ( BC) = f + g
All (ABC) = g
None (ABC) = h

Set Theory Formulas

An Example of Set Theory Question solved using Set Theory Formulas.

Question: In a class of 106 students , each student studies at least one of the three subjects Maths, Physics and Chemistry. 48 of them study Maths, 51 studies Physics and 53 Chemistry. 16 studies Maths and Physics, 17 study Maths and Chemistry and 18 study Physics and Chemistry.

Solution:
Step 1: 
To solve this set theory question by set using set theory formula first draw a venn diagram by using the details in the questions. 
Set Theory Formulas

Step 2:
Before going ahead with the set theory questions, lets find out the value of all the elements.
=> g = all(ABC) = ?
Total= A + B + C + AB - Both (AB) - Both (BC) - Both (CA) + All (ABC) + None (ABC)
By substituting the values in the above formulas,
106 = 48 + 51 + 53 + -16 - 17 - 18 + All (ABC) - o
All  ABC = 5 = g

=> e = No.of students who study chemistry and Maths - All (ABC)
e = 17 - 5 = 12

=> f = Number of students who study Physics and Chemistry - All (ABC)
f = 18 - 5 = 13

=> d = Number of students who study Maths and Physics - All (ABC)
d = 16 - 5 = 11

=> a = total number of students who study Maths - d - g - e
a = 48 - 11- 5 - 12 = 20

=> b = total number of students who study Physics - d - g - f
b = 51 - 11 - 5 - 13 = 22

=> c = total number of students who study Chemistry - e - g - f
c = 53 - 12 - 5 - 13 = 23

Question 1: The number of students who exactly study two subjects is?

Solution:
In the venn diagram  d , e and f represents number of students who studies exactly two subjects.
Number of Students who study exactly two subjects = d + e + f
Number of Students who study exactly two subjects = 11 + 12 + 13 = 36
Therefore, 36 students study exactly two subjects.

Question 2:  The number of Students who study more than two subjects?

Solution:
In the venn diagram d , e, g and f represents number of students who studies exactly more than three subjects.
Number of Students who study more than one subject = d + e + g + f
Number of Students who study more than one subject = 11 + 12 + 13 + 5 = 41
Therefore, 41 students study more than one subject.

Question 3: The number of students who study all the three subjects?

Solution:
In the venn diagram 'g' represents number of students who studies all the three subjects.
Therefore, 5 students study all the three subjects.

Question 4: The number of students who exactly study one subjects?

Solution:
In the venn diagram a, b and c represents number of students who study exactly one subject.
Number of Students who study exactly one subject = a + b + c
Number of Students who study exactly one subject = 20 + 22 + 23 = 65
Therefore, 65 students study exactly one subject.

Question 5: The number of students who study Physics and Maths but not Chemistry?

Solution:
The region that represents the students who study Physics and Maths is 'd.'
Therefore, 11 students study only Physics and Maths.

Do write in the comment section how this post had made it easy for you to solve Set Theory Questions by using Set Theory Formulas.

Stay tuned for our next post.

Set Theory Formulas
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