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!