Question

In a school there are 20 teachers who teach mathematics or physics. Of these, 12 teach mathematics and 4 teach physics and mathematics. How many teach physics?

Answer 1

Hint: We will apply the property of cardinal sets to solve this question further. This formula is represented as n(P $\cup $ Q) = n(P) + n(Q) - n(P $\cap $ Q). Here n represents numbers of the particular set.

Complete step-by-step answer:
We will consider the number of teachers who teach physics as Q and the number of teachers who teach mathematics is represented as P. From the question we see that the teachers who teach mathematics or physics are 20. Therefore we have that n(P $\cup $ Q) = 20. Out of these teachers the number of teachers who teach mathematics only is 12. So, we can write n(P) = 12. Also, we come to know that there are 4 teachers who teach physics and mathematics both. Thus we have that n(P $\cap $ Q) = 4.
Now as we have finite numbers of teachers therefore we can apply the property for cardinal sets here which is represented by n(P $\cup $ Q) = n(P) + n(Q) - n(P $\cap $ Q). By substituting the respective values to the formula we have that 20 = 12 + n(Q) - 4. By placing all the values to the right hand side of equal sign and n(Q) to the left side of equal sign we have that - n(Q) = 12 - 4 - 20 or, -n(Q) = -12. After cancelling the signs from both the sides we have that n(Q) = 12.
The Venn diagram for the question is shown below with U as a universal set.

Hence, the number of teachers who teach only physics is 12.

Note: There is a restriction to use the property of cardinal sets. That is it cannot be used for infinite sets like sets of natural numbers and so on. Do not forget to change the signs while pacing the numbers to the either side of the equal sign.