1. **State the problem:** We are given a Venn diagram with sets A and B inside a universal set E, and we need to find the number of elements in various sets. 2. **Recall definitions:** - $n(A)$ is the number of elements in set A. - $n(B)$ is the number of elements in set B. - $n(A \cap B)$ is the number of elements common to both A and B. - $n(E)$ is the total number of elements in the universal set E. - $n(A \cup B)$ is the number of elements in either A or B or both. - $n(B' \cap A)$ is the number of elements in A but not in B. 3. **Identify elements from the diagram:** - Elements in A only: b, s, o - Elements in A \cap B: a, n, r - Elements in B only: f, h - Elements outside A and B but inside E: d, w 4. **Calculate each requested value:** - $n(A) = $ elements in A only + elements in A \cap B = 3 + 3 = 6$ - $n(B) = $ elements in B only + elements in A \cap B = 2 + 3 = 5$ - $n(A \cap B) = $ elements in overlap = 3$ - $n(E) = $ all elements inside rectangle = b, s, o, a, n, r, f, h, d, w = 10$ - $n(A \cup B) = n(A) + n(B) - n(A \cap B) = 6 + 5 - 3 = 8$ - $n(B' \cap A) = $ elements in A but not in B = b, s, o = 3$ 5. **Final answers:** - a. $n(A) = 6$ - b. $n(B) = 5$ - c. $n(A \cap B) = 3$ - d. $n(E) = 10$ - e. $n(A \cup B) = 8$ - f. $n(B' \cap A) = 3$