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Let's analyze the logical statements provided and determine their truth values over the domain of real numbers.

Given statement:

\[
P(x^2 < y + 1)
\]

We have four different quantifier expressions:

1. **\(\forall x \forall y \, P(x^2 < y + 1)\)**

   This statement asserts that for all real numbers \(x\) and \(y\), \(x^2 < y + 1\) holds true. However, this cannot be true, as we can select \(x = 0\) and \(y = -1\), for which \(0^2 = 0\) and \(y + 1 = 0\), meaning \(0 < 0\) does not hold. Hence, this statement is **false**.

2. **\(\forall x \exists y \, P(x^2 < y + 1)\)**

   This statement asserts that for every real number \(x\), there exists a real number \(y\) such that \(x^2 < y + 1\). This is true because for any given \(x\), we can always choose \(y = x^2\), which satisfies \(x^2 < x^2 + 1\). Hence, this statement is **true**.

3. **\(\exists x \forall y \, P(x^2 < y + 1)\)**

   This statement asserts that there exists a real number \(x\) such that for all real numbers \(y\), \(x^2 < y + 1\). This cannot be true, as there is no fixed \(x\) such that \(x^2\) is less than every possible \(y + 1\). For instance, if we choose any fixed \(x\), we can always find a sufficiently large \(y\) such that \(x^2 \geq y + 1\). Therefore, this statement is **false**.

4. **\(\exists x \exists y \, P(x^2 < y + 1)\)**

   This statement asserts that there exists some real number \(x\) and some real number \(y\) such that \(x^2 < y + 1\). This is true because we can pick, for example, \(x = 0\) and \(y = 1\), and we get \(0^2 = 0 < 1 + 1 = 2\), which satisfies the inequality. Hence, this statement is **true**.

### Summary of Truth Values:
- (1) \(\forall x \forall y \, P(x^2 < y + 1)\): **False**
- (2) \(\forall x \exists y \, P(x^2 < y + 1)\): **True**
- (3) \(\exists x \forall y \, P(x^2 < y + 1)\): **False**
- (4) \(\exists x \exists y \, P(x^2 < y + 1)\): **True**

Would you like more details on any specific part of this analysis?

Here are five related questions that expand on this problem:
1. How would the truth values change if we restrict \(x\) and \(y\) to only positive real numbers?
2. How does the interpretation change if we replace \(x^2 < y + 1\) with \(x^2 \leq y + 1\)?
3. What happens to the truth values if we consider the domain to be integers instead of real numbers?
4. How can we verify each statement using counterexamples or specific values?
5. How do the quantifiers impact the overall truth value of logical expressions?

**Tip:** When dealing with quantifiers, look for counterexamples to quickly determine if a universally quantified statement (\(\forall\)) might be false or if an existentially quantified statement (\(\exists\)) might be true.

Tentukan nilai kebenaran pernyataan di bawah, bila domain pembicaraannya himpunan bilangan real: (a) ∀x∀y P(x² < y + 1) ∀x∃y P(x² < y + 1) ∃x∀y P(x² < y + 1) ∃x∃y P(x² < y + 1)

Determine the truth values of quantified logical statements involving real numbers for the predicate P(x² < y + 1). This problem covers fundamental concepts in predicate logic and quantifiers, suitable for undergraduate students.

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