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