Final answer:
Sets A and B and their intersection are represented on a Venn diagram with overlapping circles. The number of elements in the union of sets A and B is calculated by the formula n(A ∪ B) = n(A) + n(B) - n(A ∩ B), accounting for the shared elements between A and B.
Step-by-step explanation:
When illustrating sets with a Venn diagram, each set is represented by a circle within a larger rectangle that represents the universal set. To represent the sets A and B and their intersection, we draw two overlapping circles. The number of elements in set A, denoted n(A), is represented by the area within circle A but outside of the intersection, and it is given as x. Similarly, n(B) is represented by the area within circle B but outside of the intersection with circle A.
The intersection, n(A ∩ B), is where both circles overlap and is represented by z. To find the number of elements in the union of sets A and B, denoted n(A ∪ B), we add the numbers of elements in sets A and B and then subtract the intersection to avoid double-counting. This is due to the Principle of Inclusion-Exclusion in set theory. Therefore, n(A ∪ B) = n(A) + n(B) - n(A ∩ B), which simplifies to n(A ∪ B) = x + n(B) - z.