. Determine the truth value of each of these statements if the domain consists of all integers. a) ∀n(n + 1 > n) b) ∃n(2n = 3n) c) ∃n(n = −n) d) ∀n(3n ≤ 4n)

Let's break down and analyze the truth value of each statement given the domain consists of all integers.

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### a) \( \forall n \, (n + 1 > n) \)
- This statement reads: "For all integers \( n \), \( n + 1 > n \)."
- This is true because, for any integer \( n \), adding 1 to \( n \) always results in a number greater than \( n \).
- **Truth value: True**

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### b) \( \exists n \, (2n = 3n) \)
- This statement reads: "There exists an integer \( n \) such that \( 2n = 3n \)."
- Rearranging the equation \( 2n = 3n \), we get \( n = 0 \). So, \( n = 0 \) is the only solution.
- Since there is indeed an integer \( n = 0 \) that satisfies the equation, this statement is true.
- **Truth value: True**

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### c) \( \exists n \, (n = -n) \)
- This statement reads: "There exists an integer \( n \) such that \( n = -n \)."
- Solving \( n = -n \), we get \( 2n = 0 \), which gives \( n = 0 \).
- So, \( n = 0 \) is the only integer that satisfies this equation.
- **Truth value: True**

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### d) \( \forall n \, (3n \leq 4n) \)
- This statement reads: "For all integers \( n \), \( 3n \leq 4n \)."
- Rearranging, we get \( 0 \leq n \). This means the statement is true for all non-negative integers (\( n \geq 0 \)), but false for negative integers.
- Since the domain includes all integers, and the statement doesn't hold for negative integers, this statement is false.
- **Truth value: False**

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### Final Results:
- a) True
- b) True
- c) True
- d) False

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Would you like more details or have any other questions?

Here are 5 related questions:
1. What does the symbol \( \forall \) represent in logic?
2. How do we determine if a statement is universally true over all integers?
3. Can you think of other examples where \( \exists \) might hold for a specific integer value?
4. What would happen to the truth value in (d) if the domain were non-negative integers?
5. How would the inequality in (d) change if it were \( 3n < 4n \)?

**Tip:** When solving logical statements, carefully examine the domain (like all integers) as it significantly affects the truth of the statement!

Determine the truth value of each of these statements if the domain consists of all integers. a) ∀n(n + 1 > n) b) ∃n(2n = 3n) c) ∃n(n = −n) d) ∀n(3n ≤ 4n)

Explore the truth value of logical statements involving universal and existential quantifiers over the domain of integers. Analyze key concepts like inequalities and integer properties, with step-by-step breakdowns of statements such as ∀n(n + 1 > n) and ∃n(n = -n). Suitable for high school students in Grades 9-12.

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