The image contains two questions:

Q15: Determine the truth value of each of the following statements. (Assume the universe of discourse is the set of real numbers, $\mathbb{R}$.)

1. $\exists x (x = x + 1)$
2. $\forall x (x^2 \geq 0)$
3. $\exists x (x^2 - 201x + 5 = 0)$
4. $\exists x \forall y (x + y = 4)$

Q16: Let the universe of discourse of the proposition function $P(x, y)$ consist of the pairs $(x, y)$ where $x$ is one of $a$, $b$, or $c$, and $y$ is $a$, $b$, or $c$.

Rewrite the following propositions using disjunction and conjunction:

1. $\forall x \exists y (P(x, y))$
2. $\exists y \forall x (P(x, y))$

Solutions

Q15

1. $\exists x (x = x + 1)$:

   • This statement asserts that there exists a real number $x$ such that $x = x + 1$.
   • This is false because no real number can satisfy $x = x + 1$.

2. $\forall x (x^2 \geq 0)$:

   • This statement asserts that for all real numbers $x$, $x^2$ is greater than or equal to 0.
   • This is true because the square of any real number is always non-negative.

3. $\exists x (x^2 - 201x + 5 = 0)$:

   • This statement asserts that there exists a real number $x$ such that $x^2 - 201x + 5 = 0$.
   • Since this is a quadratic equation, it does have real solutions. Therefore, this statement is true.

4. $\exists x \forall y (x + y = 4)$:

   • This statement asserts that there exists a real number $x$ such that for all real numbers $y$, $x + y = 4$.
   • This is false because if $x$ is fixed, $x + y = 4$ cannot hold for all real $y$, as it would require $y = 4 - x$, which is only true for a specific $y$.

Q16

Given the pairs $(x, y)$ where $x$ and $y$ are each in $\{a, b, c\}$, rewrite the propositions using disjunction and conjunction.

1. $\forall x \exists y (P(x, y))$:

   • This statement means that for each $x \in \{a, b, c\}$, there exists some $y \in \{a, b, c\}$ such that $P(x, y)$ is true.
   • In disjunction and conjunction terms: $(P(a, a) \lor P(a, b) \lor P(a, c)) \land (P(b, a) \lor P(b, b) \lor P(b, c)) \land (P(c, a) \lor P(c, b) \lor P(c, c))$

2. $\exists y \forall x (P(x, y))$:

   • This statement means that there exists some $y \in \{a, b, c\}$ such that for all $x \in \{a, b, c\}, P(x, y)$ is true.
   • In disjunction and conjunction terms: $(P(a, a) \land P(b, a) \land P(c, a)) \lor (P(a, b) \land P(b, b) \land P(c, b)) \lor (P(a, c) \land P(b, c) \land P(c, c))$

Would you like more details or have any questions on these solutions?

Here are some related questions to explore further:

1. What is the difference between existential and universal quantifiers in logic?
2. How can one determine the truth value of a statement involving quantifiers?
3. How does the quadratic formula help in solving existential quantifier statements?
4. What are some examples of statements that are always false in real numbers?
5. How can logical expressions be converted into disjunctions and conjunctions effectively?

Tip: When handling quantifiers, remember that $\forall x$ requires a statement to be true for every element in the domain, while $\exists x$ only requires it to hold for at least one element.