Let's work through the questions from the image you provided.

1. Determine if the given reasoning is valid using the direct or indirect method, and justify the chosen method.

The reasoning is as follows: $\frac{q \land p, \ q \lor r, \ s \rightarrow r}{\sim s \land p}$

This is a problem in formal logic. The reasoning could be analyzed using truth tables, natural deduction, or resolution methods (direct or indirect). We aim to determine if the conclusion $\sim s \land p$ follows from the premises.

Approach:

• Direct Method: Attempt to derive $\sim s \land p$ directly from the premises.
• Indirect Method (Proof by contradiction): Assume the negation of the conclusion $\neg (\sim s \land p)$ and attempt to derive a contradiction.

To answer this correctly, we would typically check the logical flow, and based on the structure, an indirect method may be preferable (proof by contradiction).

2. Prove that the sum of two consecutive natural numbers is odd.

Let the two consecutive natural numbers be $n$ and $n+1$. The sum of these numbers is: $n + (n+1) = 2n + 1$ Since $2n + 1$ is of the form $2k + 1$, where $k = n$, the sum is always odd. Thus, the statement is true.

3. Given:

• $U = \mathbb{R}$
• $a(x): x^2 - x - 6 \leq 0$
• $b(x): 2x^2 - 6x + 23 = 0$
• $c(x): 3 \cdot |3x + 1| > 6$
• $D = (-5, 0) \cup [6, +\infty)$

We are tasked to find the truth set for:

• a) $b(x) \lor (a(x) \land c(x))$
• b) $d(x) \rightarrow a(x)$

For each of these, we need to solve the inequalities and equalities for $a(x)$, $b(x)$, and $c(x)$, and then analyze the logical expressions.

4. Given set $A = \{-4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}$, determine the truth value of the following propositions:

• a) $\forall x \in A, \exists y \in A / x = -y$
• b) $\exists x \in A, \exists y \in A / xy + 2x - 3y + 6 = 0$
• c) $\forall x \in A, \forall y \in A: (x > 0) \land ((x + y) < 10)$

This problem involves quantifiers and set theory. Each statement will need to be evaluated for the elements in the set $A$.

Would you like to explore detailed solutions for any specific question or more clarity on the logical deductions?

5 Related Questions:

1. How can we solve logical reasoning problems using truth tables?
2. What is the method of proof by contradiction, and how does it apply here?
3. Can the sum of any two even numbers be odd? Why or why not?
4. How do you solve compound inequalities involving absolute values like in $c(x)$?
5. How do quantifiers change the meaning of propositions in set theory?

Tip: When working with logical propositions, breaking the problem down using truth tables can be an effective strategy to ensure accuracy.