Question

State De-Morgan's laws. A group of 123 workers went to a canteen for cold drinks, ice-cream and tea, 42 workers took ice-cream, 36 tea and 30 cold drinks. 15 workers purchased ice-cream and tea,. 10 ice-cream and cold drinks, and 4 cold drinks and tea but not ice-cream, 11 took ice-cream and tea but not cold drinks. Determine how many workers did not purchase any things?

Answer 1

Given $N=123,n\left(I\right)=42,n\left(T\right)=36,n\left(C\right)=30,n(I\cap T)=15,n(I\cap C)=10,n(I\cap T\cap C^{\prime })=11,n(C\cap T\cap I^{\prime })=4$
Since, $n(I\cap T\cap C^{\prime })=11$
$\Rightarrow n(I\cap T)-n(I\cap T\cap C)=11$
$\Rightarrow 15-n(I\cap T\cap C)=11$
$\Rightarrow n(I\cap T\cap C)=4$
Also,$n(C\cap T\cap I^{\prime })=4$
$\Rightarrow n(C\cap T)-n(C\cap T\cap I)=4$
$\Rightarrow n(C\cap T)=8$
Now, $n(C\cup T\cup I)=n\left(C\right)+n\left(T\right)+n\left(I\right)-n(C\cap T)-n(T\cap I)-n(C\cap I)+n(C\cap T\cap I)$
$\Rightarrow n(C\cup T\cup I)=59$
Now, number of person who did not purchase anything
$n(C^{\prime }\cap T^{\prime }\cap I^{\prime })=n\left(\right(C\cup T\cup I{)}^{\prime })$
$=N-n(C\cup T\cup I)$
$=123-59=64$