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### 1. **For each pair of propositions \( P \) and \( Q \) below, state whether or not \( P \equiv Q \)**.

- **a)** \( P = p, Q = p \lor q \)

We are asked whether \( p \equiv p \lor q \). This is **false** in general because \( p \lor q \) could be true even when \( p \) is false, depending on \( q \).

- **b)** \( P = p \land q, Q = \neg p \lor \neg q \)

The expression \( P \) is \( p \land q \), which is true if both \( p \) and \( q \) are true. The expression \( Q \) is \( \neg p \lor \neg q \), which is true if at least one of \( p \) or \( q \) is false. These are negations of each other, so \( P \not\equiv Q \).

- **c)** \( P = p \to q, Q = \neg p \lor q \)

This is **true** because \( p \to q \) is logically equivalent to \( \neg p \lor q \) by the definition of implication.

- **d)** \( P = p \land (\neg q \lor r), Q = p \lor (q \land \neg r) \)

We need to check whether these two compound propositions are logically equivalent. In this case, \( P \) and \( Q \) represent two different logical expressions, so they are generally **not equivalent**.

### 2. **Determine the truth value of each statement below (domain of discourse is \( \mathbb{R} \))**.

- **a)** \( \forall x (x^2 > x) \)

This is **false**, because for \( x = 1 \), we have \( x^2 = x \), and for \( 0 < x < 1 \), \( x^2 < x \).

- **b)** \( \exists x (x^2 > x) \)

This is **true** because there exist values of \( x \), such as \( x = 2 \), for which \( x^2 > x \).

- **c)** \( \forall x (x > 1 \to x^2 > x) \)

This is **true**, because for any \( x > 1 \), we have \( x^2 > x \).

- **d)** \( \exists x (x > 1 \land x^2 > x) \)

This is **true**, because there exist values of \( x \), such as \( x = 2 \), for which \( x > 1 \) and \( x^2 > x \).

### 3. **Determine the truth value of each statement below (domain of discourse is \( \mathbb{R} \times \mathbb{R} \))**.

- **a)** \( \forall x \forall y (x^2 < y + 1) \)

This is **false**. For example, take \( x = 2 \) and \( y = 1 \). Then \( x^2 = 4 \), which is not less than \( y + 1 = 2 \).

- **b)** \( \forall x \exists y (x^2 < y + 1) \)

This is **true**, because for any \( x \), you can always find a \( y \) such that \( x^2 < y + 1 \) by choosing a sufficiently large \( y \).

- **c)** \( \exists x \forall y (x^2 < y + 1) \)

This is **false** because for any fixed \( x \), you cannot make \( x^2 \) less than \( y + 1 \) for all \( y \). For large values of \( y \), \( x^2 \geq y + 1 \).

- **d)** \( \exists x \exists y (x^2 < y + 1) \)

This is **true** because you can always find \( x \) and \( y \) such that \( x^2 < y + 1 \), for example, \( x = 1 \) and \( y = 1 \).

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Do you have any questions or would you like more details on any of these solutions?

### 5 Related Questions:
1. What does it mean for two propositions to be logically equivalent?
2. How can implication (\( \to \)) be transformed into disjunction (\( \lor \)) in logical statements?
3. Why does \( x^2 > x \) not hold for all real numbers \( x \)?
4. How can quantifiers affect the truth value of a logical statement in mathematics?
5. What strategies can be used to determine whether statements involving multiple quantifiers are true?

### Tip:
When working with logical equivalence, remember that transforming the expressions using standard logical identities (e.g., De Morgan's laws, distributive properties) can help clarify their structure.

1. For each pair of propositions P and Q below, state whether or not P ≡ Q.
  a) P = p, Q = p ∨ q
  b) P = p ∧ q, Q = ¬p ∨ ¬q
  c) P = p → q, Q = ¬p ∨ q
  d) P = p ∧ (¬q ∨ r), Q = p ∨ (q ∧ ¬r)

2. Determine the truth value of each statement below. The domain of discourse is R. Justify your answers.
  a) ∀x(x² > x)
  b) ∃x(x² > x)
  c) ∀x(x > 1 → x² > x)
  d) ∃x(x > 1 → x² > x)

3. Determine the truth value of each statement below. The domain of discourse is R × R. Justify your answers.
  a) ∀x∀y(x² < y + 1)
  b) ∀x∃y(x² < y + 1)
  c) ∃x∀y(x² < y + 1)
  d) ∃x∃y(x² < y + 1)

Analyze the logical equivalence of propositions and determine the truth value of quantified statements over real numbers. Understand core concepts in propositional logic, including logical equivalence formulas, De Morgan's Laws, and quantifier rules. Suitable for undergraduate mathematics or logic courses.

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