1. **Stating the problem:** We have a survey with respondents categorized as Men, Women, and Children, and their answers as Yes, No, and Don't know. Sets defined: - $A$ = set of adults (Men and Women) - $B$ = set of women and children - $C$ = set of "Yes" answers - $N$ = set of "No" answers We need to find: (i) $A'$ (complement of adults) (ii) $A \cap B$ (intersection of adults and women & children) (iii) $(C \cup N)'$ (complement of union of Yes and No answers) (iv) $A \cap (C \cup N)'$ 2. **Step i: Find $A'$** - $A$ = Men + Women = all adults - From the table, Men = 100 + 80 + 20 = 200 respondents - Women = 80 + 60 + 20 = 160 respondents - Total adults $|A| = 200 + 160 = 360$ - Total respondents = Men + Women + Children - Children = 50 + 20 + 30 = 100 - Total respondents = 200 + 160 + 100 = 460 - $A'$ = respondents who are not adults = Children = 100 3. **Step ii: Find $A \cap B$** - $A$ = adults (Men + Women) - $B$ = women and children - Intersection $A \cap B$ = women (since women are in both sets) - Number of women respondents = 160 4. **Step iii: Find $(C \cup N)'$** - $C$ = "Yes" answers - $N$ = "No" answers - $C \cup N$ = all respondents who answered Yes or No - From the table, total Yes = 100 + 80 + 50 = 230 - Total No = 80 + 60 + 20 = 160 - Total Yes or No = 230 + 160 = 390 - Total respondents = 460 - $(C \cup N)'$ = respondents who answered "Don't know" - From the table, Don't know = 20 + 20 + 30 = 70 5. **Step iv: Find $A \cap (C \cup N)'$** - $A$ = adults (Men + Women) = 360 - $(C \cup N)'$ = Don't know respondents = 70 - Adults who answered Don't know = Men Don't know + Women Don't know = 20 + 20 = 40 **Final answers:** (i) $A' = 100$ (ii) $A \cap B = 160$ (iii) $(C \cup N)' = 70$ (iv) $A \cap (C \cup N)' = 40$