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To construct the truth table for each of the logical statements, let’s analyze each expression using the truth values of \( p \) and \( q \).

Let:
- \( p \): True or False
- \( q \): True or False

### Truth Table Layout

We'll consider all possible combinations of \( p \) and \( q \): \( (p, q) = (T, T), (T, F), (F, T), (F, F) \).

### Expressions to Evaluate

1. **I. \( \sim(p \land q) \lor q \)**
   - This is evaluated by finding \( p \land q \), then negating it \( \sim(p \land q) \), and finally applying \( \lor \) with \( q \).

2. **II. \( p \rightarrow (p \lor q) \)**
   - This involves finding \( p \lor q \) and then evaluating the implication \( p \rightarrow (p \lor q) \).

3. **III. \( p \land (q \land \sim p) \)**
   - This requires finding \( \sim p \), then \( q \land \sim p \), and finally evaluating \( p \land (q \land \sim p) \).

4. **IV. "Exclusive OR" (often denoted as \( p \oplus q \))**
   - The exclusive OR (XOR) is true when exactly one of \( p \) or \( q \) is true, but not both.

### Truth Table

Let's construct this step-by-step.

| \( p \) | \( q \) | \( p \land q \) | \( \sim(p \land q) \) | \( \sim(p \land q) \lor q \) | \( p \lor q \) | \( p \rightarrow (p \lor q) \) | \( \sim p \) | \( q \land \sim p \) | \( p \land (q \land \sim p) \) | \( p \oplus q \) |
|---------|---------|-----------------|------------------------|-----------------------------|----------------|-------------------------------|-------------|-----------------------|-------------------------------|----------------|
| T       | T       | T               | F                      | T                           | T              | T                             | F           | F                     | F                             | F              |
| T       | F       | F               | T                      | T                           | T              | T                             | F           | F                     | F                             | T              |
| F       | T       | F               | T                      | T                           | T              | T                             | T           | T                     | F                             | T              |
| F       | F       | F               | T                      | F                           | F              | T                             | T           | F                     | F                             | F              |

### Results for Each Expression

- **I.** \( \sim(p \land q) \lor q \): Column \( \sim(p \land q) \lor q \)
- **II.** \( p \rightarrow (p \lor q) \): Column \( p \rightarrow (p \lor q) \)
- **III.** \( p \land (q \land \sim p) \): Column \( p \land (q \land \sim p) \)
- **IV.** Exclusive OR (\( p \oplus q \)): Column \( p \oplus q \)

Would you like further explanations or have any questions?

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### Related Questions

1. How does the negation operator affect a conjunction or disjunction?
2. How do we interpret the implication \( p \rightarrow q \) in truth tables?
3. Can we generalize exclusive OR to more than two variables?
4. How do we simplify logical expressions using identities?
5. What are common pitfalls in constructing truth tables?

### Tip

When constructing truth tables, it’s useful to break down complex expressions into smaller components. This allows each sub-expression’s truth value to be calculated step-by-step, ensuring accuracy.

Construct the truth table of the following: ~ (p ∧ q) ∨ q, p → (p ∨ q), p ∧ (q ∧ ~p), and Exclusive OR.

This page provides a detailed solution for constructing truth tables for various logical expressions, including negation, conjunction, disjunction, implication, and exclusive OR. Ideal for high school students in Grades 11-12 studying propositional logic.

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