Let us analyze each statement in the context of the domain of integers to determine which are true:

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A. ∃n such that n² ≤ −1

• Analysis: The square of any integer $n$ is $n^2$, which is always non-negative ($n^2 \geq 0$). Thus, there cannot exist an integer $n$ such that $n^2 \leq -1$, as $n^2 \geq 0$ for all integers.

• Verdict: False

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B. ∀n, n² ≥ 0

• Analysis: The square of any integer $n$ is always non-negative ($n^2 \geq 0$) because multiplying any number by itself yields a positive or zero result. This is true for all integers.

• Verdict: True

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C. ∃n such that n² = 2

• Analysis: There is no integer $n$ such that $n^2 = 2$. The square root of 2 is irrational and cannot be represented as an integer.

• Verdict: False

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D. ∀n, n² ≥ n

• Analysis: For $n \geq 1$, $n^2 \geq n$ holds true. However, for integers $n$ such as 0 and $n < 0$, the inequality $n^2 \geq n$ also holds:

  • $n = 0$: $n^2 = 0 \geq 0$
  • $n = -1$: $n^2 = 1 \geq -1$
  • For all negative integers, $n^2$ is positive and thus $n^2 \geq n$ holds.

• Verdict: True

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Final Answer:

• The true statements are: B and D.

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Follow-up Questions:

1. Why is $n^2$ always non-negative for all integers?
2. How can we prove that $\sqrt{2}$ is irrational and thus not an integer?
3. Can you find a counterexample for $n^2 \geq n$ among integers?
4. What is the general behavior of $n^2$ compared to $n$ for negative, zero, and positive integers?
5. What are some similar inequalities involving squares of integers that can be analyzed?

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Tip: Always check the nature of a mathematical operation (e.g., squaring) to determine its range of possible values in a given domain.