VerifiedHint: We will apply the formula of the cardinal property of the given sets. Since, the sets are finite sets, we can use here the formula given by 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 first consider that the people who like coffee are represented as P and the people who like tea will be denoted by Q.
According to the question, we have a group of 70 people. That is the total number of people is 70 and this can be written as n(P $\cup $ Q) = 70.
We are given that the number of people who like coffee is 37. So, we actually have n(P) = 37. Moreover we have a number of people who like to drink tea. Therefore, we can write it as n(Q) = 52.
We are supposed to find out the number of people who like both coffee and tea. That is, we need to find the value of n(P $\cap $ Q). Since all the values are finite values therefore we will apply here the formula of cardinal property which can be written as n(P $\cup $ Q) = n(P) + n(Q) - n(P $\cap $ Q). Here n represents numbers of the particular set.
After substituting the values we have 70 = 37 + 52 - n(P $\cap $ Q). By taking all the numbers to the left hand side of the equal sign we have 70 - 37 - 52 = - n(P $\cap $ Q).
Therefore the value of - n(P $\cap $ Q) is -19. Or by cancelling the minus sign from both the sides we have n(P $\cap $ Q) = 19.
The Venn diagram for the question is shown below along with U representing a universal set.
Hence, the number of people who like to drink both coffee and tea is 19.
Note: While using the words to form union and intersection one should be cleared with the terms of it. For example we use intersection whenever we are given the word ‘and’ in the question. Also, if we are given the word ‘or’ then we use it as a union sign or operation.
