In a school there are $20$ teachers who teach mathematics or physics. Of these, $12$ teach mathematics and $4$ teach both physics and mathematics. How many teach physics ?
In a school there are $20$ teachers who teach mathematics or physics. Of these, $12$ teach mathematics and $4$ teach both physics and mathematics. How many teach physics ?
Let $M$ denote the set of teachers who teach mathematics and $P$ denote the set of teachers who teach physics. In the statement of the problem, the word 'or' gives us a clue of union and the word 'and' gives us a clue of intersection. We, therefore, have
$n( M \cup P )=20, n( M )=12 \text { and } n( M \cap P )=4$
We wish to determine $n( P ).$
Using the result $n( M \cup P )=n( M )+n( P )-n( M \cap P )$
we obtain $20=12+n(P)-4$
Thus $n( P )=12$
Hence $12$ teachers teach physics.
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