### Theory:

In a survey of $$200$$ people, it is observed that $$125$$ of them use Television, $$120$$ of them use Radio and $$135$$ of them use Mobile. But, it is noted that $$70$$ of them use both Television and Radio, $$85$$ of them use both Radio and Mobile, $$75$$ of them use both Television and Mobile, and $$50$$ of them use all the three devices. Find the following using the Venn diagram.

(i) How many of them use only Television and Radio, not Mobile?

(ii) How many of them use only Radio and Mobile, not Television?

(iii) How many of them use only Television and Mobile, not Radio?

(iv) How many of them use only Television?

(v) How many of them use only Radio?

(vi) How many of them use only Mobile?

Solution:

Let us say, $$A$$, $$B$$ and $$C$$ represent the sets of people who use Television, Radio, and Mobile, respectively.

Total number of people $$= n(U) = 200$$

Number of people use Television $$= n(A) = 125$$

Number of people use Radio $$= n(B) = 120$$

Number people use Mobile $$= n(C) = 135$$

Number of people use Television and Radio $$= n(A \cap B) = 70$$

Number of people use Radio and Mobile $$= n(B \cap C) = 85$$

Number of people use Television and Mobile $$= n(A \cap C) = 75$$

Number of people use all the three $$= n(A \cap B \cap C) = 50$$ From the diagram:

(i) Number of people use only Television and Radio, but not a Mobile $$= 20$$

(ii) Number of people use only Radio and Mobile, but not a Television $$= 35$$

(iii) Number of people use only Television and Mobile, but not a Radio $$= 25$$

(iv) Number of people use only Television $$= 30$$

(v) Number of people use only Radio $$= 15$$

(vi) Number of people use only Mobile $$= 25$$