1. **Problem Statement:** We have a table of 231 employees classified by job category (B1 to B6) and age category (A1 to A5). We need to explain and find the number of employees in various set intersections and unions, and answer specific questions about employee counts based on given conditions. 2. **Understanding the Table:** The table shows counts for each job and age category intersection. For example, B1 ∩ A5 means clerical employees older than 35. 3. **Set Notation and Counting:** - Intersection $A \cap B$ means employees in both sets. - Union $A \cup B$ means employees in either set or both. 4. **Calculations for each part:** (a) $B1 \cap A5$: Clerical employees older than 35. From table: 5 employees. (b) $A2 \cap B6$: Executives aged 21-25. From table: 0 employees. (c) $B4 \cap A5$: Salespeople older than 35. From table: 2 employees. (d) $A1 \cup B6$: Employees aged ≤ 20 or executives. Calculate $n(A1) = 39$ and $n(B6) = 5$. Intersection $A1 \cap B6$ is executives aged ≤ 20: from table 0. So, $$n(A1 \cup B6) = n(A1) + n(B6) - n(A1 \cap B6) = 39 + 5 - 0 = 44$$ (e) $A3 \cup A5$: Employees aged 26-30 or older than 35. Calculate $n(A3) = 70$, $n(A5) = 19$. Intersection $A3 \cap A5 = \emptyset$ (age categories are disjoint). So, $$n(A3 \cup A5) = 70 + 19 = 89$$ (f) $B2 \cup B5$: Custodial or junior executives. Calculate $n(B2) = 15$, $n(B5) = 8$. Intersection $B2 \cap B5 = \emptyset$ (different job categories). So, $$n(B2 \cup B5) = 15 + 8 = 23$$ (g) $A4$: Employees aged 31-35. From table total: 41 employees. (h) $(A1 \cup A2) \cap B5$: Junior executives aged ≤ 25. Calculate $n(A1 \cap B5) = 0$, $n(A2 \cap B5) = 1$. So, $$n((A1 \cup A2) \cap B5) = 0 + 1 = 1$$ (i) $(B3 \cup B4) \cap A5$: Craft workers or salespeople older than 35. Calculate $n(B3 \cap A5) = 10$, $n(B4 \cap A5) = 2$. Intersection $B3 \cap B4 = \emptyset$. So, $$n((B3 \cup B4) \cap A5) = 10 + 2 = 12$$ 5. **Additional Conditions:** (i) Neither executive nor junior executive means not in $B5$ or $B6$. Calculate total employees: 231. Calculate $n(B5) = 8$, $n(B6) = 5$. So, $$n(\text{neither}) = 231 - (8 + 5) = 218$$ (k) Both executive and junior executive means intersection $B5 \cap B6$ which is empty. So, $$n = 0$$ (l) More than 30 years old and clerical or custodial means age categories $A4$ and $A5$ with job categories $B1$ or $B2$. Calculate: $$n(B1 \cap A4) = 10, \quad n(B1 \cap A5) = 5$$ $$n(B2 \cap A4) = 2, \quad n(B2 \cap A5) = 1$$ Sum: $$10 + 5 + 2 + 1 = 18$$ (m) Salesperson and/or between 21 and 25 years old means $B4 \cup A2$. Calculate $n(B4) = 23$, $n(A2) = 62$. Intersection $B4 \cap A2 = 5$. So, $$n(B4 \cup A2) = 23 + 62 - 5 = 80$$ (n) Craft worker 35 years old or younger means $B3$ and age categories $A1$, $A2$, $A3$, $A4$. Calculate: $$15 + 30 + 35 + 20 = 100$$ (o) Craft worker or salesperson and between 21 and 30 years old means $(B3 \cup B4) \cap (A2 \cup A3)$. Calculate: $$n(B3 \cap A2) = 30, \quad n(B3 \cap A3) = 35$$ $$n(B4 \cap A2) = 5, \quad n(B4 \cap A3) = 10$$ Sum: $$30 + 35 + 5 + 10 = 80$$ (p) Clerical or custodial and more than 30 years old means $(B1 \cup B2) \cap (A4 \cup A5)$. Calculate: $$n(B1 \cap A4) = 10, \quad n(B1 \cap A5) = 5$$ $$n(B2 \cap A4) = 2, \quad n(B2 \cap A5) = 1$$ Sum: $$10 + 5 + 2 + 1 = 18$$ **Final answers:** (a) 5 (b) 0 (c) 2 (d) 44 (e) 89 (f) 23 (g) 41 (h) 1 (i) 12 (i) 218 (k) 0 (l) 18 (m) 80 (n) 100 (o) 80 (p) 18