Sets and Unions ( Venn diagram) problems - concept and short cut tips

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Questions on sets and unions are important in every bank exam. Many students fail to solve them correctly due to lack of clarity. So, today we will explain you on how to solve these problems. To make you understand the concept easily, we will go for live example and explain concept .

Example:
In a group of students, 21 students like Potato , 26 students like Tomato , 29 students like Carrot , 15 students like tomato and carrot, 14

students like tomato and potato, 12 students like potato and carrot. and 8 students like all three vegetables.
Questions:
1. How many students are there in group ?
a) 49 b) 45 c) 48 d) 43 e) 44

2. How many students do like carrot only ?
a) 12 b) 9 c) 8 d) 10

3. how many students like tomato only ?
a)5 b) 4 c) 7 d)6

Solution:

Let A , B and C be the sets of students who like Potato , Tomato and Carrot respectively . So if we represent above given information in the form of sets
n(A)= 21
n(B)= 26
n(C)= 29
n(A∩B)= 14 (Those who like both potato and tomato)
n(A∩C)= 12 ( Those who like both potato and carrot)
n(B∩ C)=15 ( Those who like both tomato and carrot)
n(A∩B∩C)=8 ( Those who like potato , tomato and carrot )

We have to determine the total no of persons in the group by using above given information before proceeding to Step 1

According to standard formula
Total no of students in group n (A U B U C) = (n (A) + n (B) + n(c)+ n(A∩B∩C)) -( n(A∩B) + n(A∩C)+(B∩C))
= (21+26+29+8) -(14+12+15)
=84-41
= 43.

Step 1: First draw 3 circles as we are given 3 classes of information.( i.e Potato, Tomato, Carrot) and 3 should be drawn in such a way that they intersect to some extent to represent the given information.

Step 2: As 8 students like all three vegetables , they should be placed at a intersection or union where all three circles meet.

Step 3: Similarly the Number of people who like Potato and Tomato , Tomato and carrot , potato and carrot are filled in the respective intersection . But in this case while filling the value we have subtract 8 from given value as 8 holds common for all the three combinations . For example students who like potato and tomato are given as 14 numbers. while filling the intersection we should use 14-8= 6 as the value.

Step 4 (Final step) : Final step is we have fill the individual classes . as given in question 21 students like potato . so we have subtract all the values which are in circle of potato from 21 . So i t becomes 21-18= 3. Similarly Tomato likers = 26-21=5 and Carrot likers = 29-19=10

Question and Answers :

1. How many students are there in group ?

a) 49 b) 45 c) 48 d) 43 e) 44

Answer: As we have determined the total number of students in the beginning , the answer is 43 (d)

2. How many students do like carrot only ?
a) 12 b) 9 c) 8 d) 10

Answer: As we can see in last step diagram, 10 students like carrot only . So answer is (d)

3. 3. how many students like tomato only ?
a)5 b) 4 c) 7 d)6

Answer: As we can see from step 4 diagram , 5 people like tomato only . So answer is (a)

4. How many students like Potato and Carrot only ?
a) 7 b)4 c) 3 d) 9

Answer: AS we can see in step 4 diagram 8 people like all three vegetables and 4 people like only potato and carrot. So answer is 4 (b)