Question

Let the Universal Set be S. Let A and B are subsets of S. Set A contains 24 elements and Set B contains 16 elements. Sets A and B have 12 elements in common. If there are 42 elements that are in S but not in A nor B, how many elements are in S?

Answer 1

Firstly, let us find out the number of unique elements in Set A and Set B.
To find the number of unique elements in Set A, we take the total number of elements in Set A, which is 24, and subtract the number of elements that Set A and B have in common, which is 12. Therefore, we have 24 - 12 = 12 unique elements in Set A.

Similarly, to find the number of unique elements in set B, we take the total number of elements in Set B, which is 16, and subtract the number of elements that Set A and B have in common, which is 12. Therefore, we have 16 - 12 = 4 unique elements in Set B.

Now, we calculate the total number of elements in the universal set S.
The total number of elements in a set is the addition of unique elements in Set A, unique elements in Set B, common elements of A and B, and elements in S that are not in both A and B. We already found the number of unique elements in Set A to be 12 and Set B to be 4. And it is given that the number of common elements of A and B (12), as well as there are 42 elements in S but not in either A or B.

Therefore, the total number of elements in S is 12 (unique A) + 4 (unique B) + 12 (common AB) + 42 (S but not AB) = 70 elements.

So, the set S has 70 elements.