1. The problem involves finding the cardinality $n(X \times Y)$ and related values from given cardinalities. 2. Recall the rule for Cartesian products cardinalities: $$n(X \times Y) = n(X) \times n(Y)$$. 3. Problem 1: Given $n(X) = 3$, $Y = \{4, 5\}$ so $n(Y) = 2$. Calculate: $$n(X \times Y) = 3 \times 2 = 6$$. Answer: (b) 6 4. Problem 2: Given $n(X) = 3$ and $n(X \times Y) = 12$, find $n(Y)$: $$n(Y) = \frac{n(X \times Y)}{n(X)} = \frac{12}{3} = 4$$. Answer: (a) 4 5. Problem 3: Given $n(X) = 5$, $n(X \times Y) = 10$, find $n(Y)$: $$n(Y) = \frac{10}{5} = 2$$. Answer: (c) 2 6. Problem 4: Given $n(X) = 5$, $n(X \times Y) = 15$, find $n(Y)$: $$n(Y) = \frac{15}{5} = 3$$. Answer: (a) 3 7. Problem 5: Given $n(X^2) = 4$, and $n(X \times Y) = 6$, find $n(Y)$. Since $X^2 = X \times X$, so $$n(X) = \sqrt{n(X^2)} = \sqrt{4} = 2$$. Then $$n(Y) = \frac{n(X \times Y)}{n(X)} = \frac{6}{2} = 3$$. Answer: (b) 3 8. Problem 6: Given $X \times Y = \{(2,3), (2,4)\}$, find $n(X)$. Elements show $x=2$ only, so one unique $x$. Answer: (b) 1 9. Problem 7: Given $X = \{2,3,4\}$, find $n(X^2) = n(X \times X)$: $$n(X^2) = n(X)^2 = 3^2 = 9$$. Answer: (c) 9 10. Problem 8: Given $X=\{7\}$, find $n(X^2)$: $$n(X) = 1 \implies n(X^2) = 1^2 = 1$$. Answer: (a) 1 11. Problem 9: Given $n(X) = 2$, $n(Y \times X) = 6$, find $n(Y^2)$. Note $n(Y \times X) = n(Y) \times n(X) = 6$, so $$n(Y) = \frac{6}{2} = 3$$. Then $$n(Y^2) = n(Y)^2 = 3^2 = 9$$. Answer: (b) 9 12. Problem 10: Given $n(X) = 2$, $n(X \times Y) = 8$, find $n(Y^2)$. $$n(Y) = \frac{8}{2} = 4$$ $$n(Y^2) = 4^2 = 16$$. Answer: (d) 16 13. Problem 11: Given $n(X \times Y) = 6$, $n(Y) = 2$, find $n(X^2)$. $$n(X) = \frac{6}{2} = 3$$ $$n(X^2) = 3^2 = 9$$. Answer: (b) 9 14. Problem 12: Given $n(X^2) = 9$, $n(X \times Y) = 6$, find $n(Y^2)$. $$n(X) = \sqrt{9} = 3$$ $$n(Y) = \frac{6}{3} = 2$$ $$n(Y^2) = 2^2 = 4$$. Answer: (d) 4 15. Problem 13: Given $X \times Y = \{(1,2), (1,3), (1,4)\}$, find $n(X) + n(Y^2)$. Unique $X$ values: 1 element, so $n(X) = 1$. Unique $Y$ values: 3 elements so $$n(Y) = 3 \implies n(Y^2) = 3^2 = 9$$ Sum: $$1 + 9 = 10$$. Answer: (d) 10 16. Problem 14: Given $X=\{2\}$, $Y=\{3\}$, find $X \times Y$. Cartesian product is a set of ordered pairs: $$\{(2, 3)\}$$. Answer: (d) {(2 , 3)}