Question

If P and Q are two sets such that P has 40 elements, P $\cup $ Q has 60 elements and P $\cap $ Q has 10 elements. How many elements does Q have?

Answer 1

Hint: We will use the value of n(P $\cup $ Q) to be equal to the number of the individual sets P and Q and subtract the value of n(P $\cap $ Q) from it. Thus can be numerically written as n(P $\cup $ Q) = n(P) + n(Q) - n(P $\cap $ Q) where n is the number here.

Complete step-by-step answer:
From the question we are clearly given the number of elements in P as 40 which means that n(P) = 40. Also we are given that the number of elements in P $\cup $ Q has 60 elements that is n(P $\cup $ Q) = 60. And P $\cap $ Q contains 10 elements therefore we have n(P $\cap $ Q) = 10.
We are asked to find the value of n(Q). It can be found out by using the numerical written as n(P $\cup $ Q) = n(P) + n(Q) - n(P $\cap $ Q) where n is the number. By substituting the respected values in the formulas we have 60 = 40 + n(Q) - 10. At this step we will place all the constants to the right side of the equal sign and the term n(Q) to the left hand side of the equal sign.
Therefore, we have that - n(Q) = 40 - 60 - 10. After solving we get - n(Q) = - 30. By cancelling the negative sign we have n(Q) = 30.
The Venn diagram for the question is shown below with U as a universal set.

Hence, the number of Q is 30.

Note: While placing the values to the either side of equal sign we have to change signs. We will not use the formula n(P $\cup $ Q) = n(P) + n(Q) - n(P $\cap $ Q) if the sets were given to us as infinite. n(P $\cap $ Q) is always negative in the formula. One should be aware not to write it as positive in the formula.