1.Set TheoryMedium

In a survey of $60$ people, it was found that $25$ people read newspaper $H , 26$ read newspaper $T, 26$ read newspaper $I, 9$ read both $H$ and $I, 11$ read both $H$ and $T,$ $8$ read both $T$ and $1,3$ read all three newspapers. Find:

the number of people who read exactly one newspaper.

Solution

Let $A$ be the set of people who read newspaper $H.$

Let $B$ be the of people who read newspaper $T.$

Let $C$ be the set of people who read newspaper $I.$

Accordingly, $n(A)=25, n(B)=26,$ and $n(C)=26$

$n(A \cap C)=9, n(A \cap B)=11,$ and $n(B \cap C)=8$

$n(A \cap B \cap C)=3$

Let $U$ be the set of people who took part in the survey.

Let $a$ be the number of people who read newspapers $H$ and $T$ only.

Let $b$ denote the number of people who read newspapers $I$ and $H$ only.

Let $c$ denote the number of people who read newspapers $T$ and $I$ only.

Let $d$ denote the number of people who read all three newspapers.

Accordingly, $d=n(A \cap B \cap C)=3$

Now, $n(A \cap B)=a+d$

$n(B \cap C)=c+d$

$n(C \cap A)=b+d$

$\therefore a+d+c+d+b+d=11+8+9=28$

$\Rightarrow a+b+c+d=28-2 d=28-6=22$

Hence, $(52-22)=30$ people read exactly one newspaper.

There are $200$ individuals with a skin disorder, $120$ had been exposed to the chemical $C _{1}, 50$ to chemical $C _{2},$ and $30$ to both the chemicals $C _{1}$ and $C _{2} .$ Find the number of individuals exposed to

Chemical $C_{2}$ but not chemical $C_{1}$

Medium

Easy

Medium

Difficult

Difficult