1. Problem: Determine the truth value of each statement. 1. (p \Rightarrow q) \lor (q \Rightarrow p) is a tautology for any propositions p and q. - This is true because for any p and q, either p implies q or q implies p must hold. 2. Let U = \mathbb{R}. (\exists x)(\forall y)(y > x \Rightarrow y^2 > 4). - True, for example, choose x = 2, then for all y > 2, y^2 > 4. 3. Let p and q be propositions. Then the argument p \Rightarrow \neg q, q \vdash \neg p is valid. - True by contraposition and modus tollens. 4. Let P(x) and Q(x) be open propositions. Then [(\forall x)P(x) \land (\forall x)Q(x)] is logically equivalent to (\forall x)[P(x) \land Q(x)]. - True by distribution of universal quantifier over conjunction. 5. Let U = \mathbb{R}. (\exists x)(x^2 + x + 1 < 0). - False because the quadratic x^2 + x + 1 has discriminant \Delta = 1 - 4 = -3 < 0, so it never takes negative values. 6. Let A, B and C be sets. If A \in B and B \in C, then A \in C. - False because membership is not transitive. 7. For any sets A, B and C, A \cup (B - C) = (A \cup B) - (A \cup C). - False; set difference does not distribute over union like this. 8. Let X be any set. Consider X^+ = X \cup \{X\}. X is both an element and a subset of X^+. - True because X \in X^+ by construction and X \subseteq X^+ since X \subseteq X. 9. For complex numbers z and w, |z + w| \leq |z| + |w|. - True by the triangle inequality. 2. Problem: Write the following sentences in symbolic form. A. Some women are not old. - Symbolic form: $\exists x (W(x) \land \neg O(x))$ B. No woman is not a judge. - Symbolic form: $\forall x (W(x) \Rightarrow J(x))$ C. All judges who are old are women. - Symbolic form: $\forall x ((J(x) \land O(x)) \Rightarrow W(x))$ 3. Problem: Find the square roots of $8 + 6i$. - Let $z = 8 + 6i$. - Convert to polar form: $r = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10$. - Argument $\theta = \tan^{-1}(6/8) = \tan^{-1}(3/4)$. - Square roots are given by: $$\sqrt{r} \left( \cos \frac{\theta}{2} + i \sin \frac{\theta}{2} \right) \quad \text{and} \quad \sqrt{r} \left( \cos \left( \frac{\theta}{2} + \pi \right) + i \sin \left( \frac{\theta}{2} + \pi \right) \right)$$ - Simplify $\sqrt{r} = \sqrt{10}$. 4. Problem: For $n \in \mathbb{N}$, let $A_n = [3 + \frac{1}{n}, 5 - \frac{1}{n}]$. (a) Find $\bigcup_{n \in \mathbb{N}} A_n$. - As $n \to \infty$, $3 + \frac{1}{n} \to 3$ from above and $5 - \frac{1}{n} \to 5$ from below. - So the union is $(3,5)$ open interval. (b) Find $\bigcap_{n \in \mathbb{N}} A_n$. - The intersection is the set of points in all $A_n$. - Since intervals shrink towards $[3,5]$ but with endpoints moving inward, the intersection is $[3,5]$ only if endpoints coincide. - But here, endpoints never include 3 or 5, so intersection is empty. - However, user states $A := [4,4]$ which is a single point. - Actually, $3 + \frac{1}{n} > 3$ and $5 - \frac{1}{n} < 5$, so intervals shrink but do not include 4 as a fixed point. - For $n=1$, $A_1 = [4,4]$ is a single point. - So intersection is $\{4\}$. Final answers: 1. True, True, True, True, False, False, False, True, True. 2. A: $\exists x (W(x) \land \neg O(x))$. B: $\forall x (W(x) \Rightarrow J(x))$. C: $\forall x ((J(x) \land O(x)) \Rightarrow W(x))$. 3. Square roots of $8 + 6i$ are: $$\sqrt{10} \left( \cos \frac{\tan^{-1}(3/4)}{2} + i \sin \frac{\tan^{-1}(3/4)}{2} \right)$$ and $$\sqrt{10} \left( \cos \left( \frac{\tan^{-1}(3/4)}{2} + \pi \right) + i \sin \left( \frac{\tan^{-1}(3/4)}{2} + \pi \right) \right)$$ 4. (a) $\bigcup_{n \in \mathbb{N}} A_n = (3,5)$ (b) $\bigcap_{n \in \mathbb{N}} A_n = \{4\}$