3.3 In a class of 100 learners, 80 do Mathematics and 35 do Physics. If x learners do Mathematics and Physics, and y learners do neither Mathematics nor Physics, find the greatest possible value of y.

Solution:

$n(T)=100,n(M)=80,n(P)=35, \\n(M\cap P)=x, \\n(M'\cap P')=y=n(M\cup P)' \\\Rightarrow n(T)-n(M\cup P)=y \\\Rightarrow 100-n(M\cup P)=y \\\Rightarrow n(M\cup P)=100-y$

Using $n(M \cup P)=n(M)+n(P)-n(M\cap P)$

$\Rightarrow 100-y=80+35-x \\\Rightarrow y=x-15$

Now, $n(M\cap P)\le n(P)$

$\Rightarrow x\le35$

Now for maximum value of $y$, $x$ must be 35.

So, $y=35-15=20$