1. **Problem statement:** Given the employee classification table by job category (B1 to B4) and age category (A1 to A5), answer the set membership and counting questions. 2. **Recall set notation:** - $B_i \cap A_j$ means employees in job category $B_i$ and age category $A_j$. - $A_i \cup B_j$ means employees in age category $A_i$ or job category $B_j$ (or both). 3. **Given table:** | Job \ Age | A1(\leq 20) | A2(21-25) | A3(26-30) | A4(31-35) | A5(>35) | Total | |-----------|-------------|-----------|-----------|-----------|---------|-------| | B1 Clerical | 20 | 20 | 15 | 10 | 5 | 70 | | B2 Custodial | 3 | 6 | 3 | 2 | 1 | 15 | | B3 Craft workers | 15 | 30 | 35 | 20 | 10 | 110 | | B4 Salespeople | 1 | 5 | 10 | 4 | 2 | 23 | 4. **Answering each part:** (a) $B1 \cap A5$ = Clerical employees older than 35 = 5 (b) $A2 \cap B6$ = Job category B6 does not exist, so number = 0 (c) $B4 \cap A5$ = Salespeople older than 35 = 2 (d) $A1 \cup B6$ = B6 does not exist, so this is just $A1$ (age ≤20) employees total: $20 + 3 + 15 + 1 = 39$ (e) $A3 \cup A5$ = Employees aged 26-30 or older than 35: Sum all employees in columns A3 and A5: $15+3+35+10$ (A3) + $5+1+10+2$ (A5) = $(15+3+35+10) + (5+1+10+2) = 63 + 18 = 81$ (f) $B2 \cup B3$ = Custodial or Craft workers: Sum totals of B2 and B3: $15 + 110 = 125$ (g) $A4$ = Employees aged 31-35: Sum column A4: $10 + 2 + 20 + 4 = 36$ (h) $(A1 \cup A2) \cap B3$ = Craft workers aged ≤25: Sum B3 employees in A1 and A2: $15 + 30 = 45$ (i) $(B1 \cup B4) \cap A5$ = Employees older than 35 who are Clerical or Salespeople: Sum B1 and B4 in A5: $5 + 2 = 7$ 5. **Additional conditions:** (i) Neither executive nor junior executive: Given only B1 to B4, and no executives or junior executives explicitly, assume none, so number = 0 (k) Both executive and junior executive: No such category, number = 0 (l) More than 30 years old and clerical or custodial: Age >30 means A4 and A5 columns. Clerical (B1) in A4 and A5: $10 + 5 = 15$ Custodial (B2) in A4 and A5: $2 + 1 = 3$ Total = $15 + 3 = 18$ (m) Salesperson and/or between 21 and 25 years old: Salespeople total = 23 Between 21 and 25 (A2) total = sum column A2: $20 + 6 + 30 + 5 = 61$ Union count = Salespeople + A2 - Salespeople in A2 Salespeople in A2 = 5 So total = $23 + 61 - 5 = 79$ (n) Craft worker 35 years old or younger: Age ≤35 means A1 to A4 columns. Craft workers in A1 to A4: $15 + 30 + 35 + 20 = 100$ (o) Craft worker or salesperson and between 21 and 30 years old: Age 21-30 means A2 and A3 columns. Craft workers in A2 and A3: $30 + 35 = 65$ Salespeople in A2 and A3: $5 + 10 = 15$ Union = 65 + 15 - overlap (none, different job categories) Total = 80 (p) Clerical or custodial and more than 30 years old: Age >30 means A4 and A5 columns. Clerical in A4 and A5: $10 + 5 = 15$ Custodial in A4 and A5: $2 + 1 = 3$ Union = $15 + 3 = 18$ **Final answers:** (a) 5 (b) 0 (c) 2 (d) 39 (e) 81 (f) 125 (g) 36 (h) 45 (i) 7 (i) 0 (k) 0 (l) 18 (m) 79 (n) 100 (o) 80 (p) 18