**Solving Word Problems Involving Sets :**

In this section, we will see how to solve word problems in set theory.

**Example 1 :**

There are 40 students in a class. 20 take Chemistry and 25 take French. 8 students take both.

(i) Find how many students take none.

(ii) How many are there in at least one classes?

**Solution :**

Total number of students = 40

Let A and B be the students who took Chemistry and French respectively.

n(A) = 20, n(B) = 25 and n(AnB) = 8

n(AUB) = n(A) + n(B) - n(AnB)

n(AUB) = 20 + 25 - 8

= 45 - 8

n(AUB) = 37

Number of students who take none = Total number of students - Number of students who take one

= 40 - 37

= 3

Hence the number of students who take none is 3 and number of students are in atleast one class is 37.

**Example 2 :**

A group of 20 college students were asked whether they are using the social net working sites. For that 8 said they use face book. 10 said they use twitter. 4 said they use both.

(i) Find how many are not using none?

(ii) Find how many are using only face book and twitter?

**Solution :**

Let A and B be the students who uses Face book and twitter respectively.

Number of students = 20

n(A) = 8, n(B) = 10, n(A n B) = 4

(i) To find the number of students who are not using any of these, we have to subtract number of students who uses one from the total number of students.

n(A U B) = n(A) + n(B) - n(A n B)

= 8 + 10 - 4

= 14

Number of students who use none = 20 - 14

= 6

Number of students who use face book = n(A) - n(A n B)

= 8 - 4

= 4

Number of students who use twitter = n(B) - n(A n B)

= 10 - 4

= 6

**Example 3 :**

At a breakfast buffet, 20 people chose coffee and 17 chose juice. 10 people chose both coffee and juice. If each person chose at least one of these beverages, how many people visited the buffet?

**Solution :**

Let A and B be the students who choose coffee and juice respectively.

n(A) = 20, n(B) = 17, n(AnB) = 10

n(AUB) = n(A) + n(B) - n(AnB)

n(AUB) = 20 + 17 - 10

= 37 - 10

n(AUB) = 27

Hence the number of people who visited the buffet is 27.

After having gone through the stuff given above, we hope that the students would have understood, how to solve word problems involving sets.

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