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To construct a truth table for the expression \( p \rightarrow (q \lor \sim p) \), let's break down each component step-by-step and evaluate the expression for all possible truth values of \( p \) and \( q \).

### Step 1: Identify the Components
The expression \( p \rightarrow (q \lor \sim p) \) has three parts:
1. \( p \)
2. \( q \)
3. \( \sim p \) (the negation of \( p \))

### Step 2: Construct the Inner Expression \( q \lor \sim p \)
The part inside the parentheses, \( q \lor \sim p \), represents a logical "OR" operation between \( q \) and the negation of \( p \).

### Step 3: Evaluate \( p \rightarrow (q \lor \sim p) \)
The outermost part, \( p \rightarrow (q \lor \sim p) \), is a conditional ("if-then") statement.

### Truth Table
Let's fill in the truth values step-by-step for all four possible combinations of \( p \) and \( q \).

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

### Explanation of the Truth Table Rows
1. **Row 1**: \( p = T \), \( q = T \)
   - \( \sim p = F \)
   - \( q \lor \sim p = T \lor F = T \)
   - \( p \rightarrow (q \lor \sim p) = T \rightarrow T = T \)

2. **Row 2**: \( p = T \), \( q = F \)
   - \( \sim p = F \)
   - \( q \lor \sim p = F \lor F = F \)
   - \( p \rightarrow (q \lor \sim p) = T \rightarrow F = F \)

3. **Row 3**: \( p = F \), \( q = T \)
   - \( \sim p = T \)
   - \( q \lor \sim p = T \lor T = T \)
   - \( p \rightarrow (q \lor \sim p) = F \rightarrow T = T \)

4. **Row 4**: \( p = F \), \( q = F \)
   - \( \sim p = T \)
   - \( q \lor \sim p = F \lor T = T \)
   - \( p \rightarrow (q \lor \sim p) = F \rightarrow T = T \)

Would you like further explanation on each step?

### Related Questions:
1. What are the truth values for other logical operators like "AND" and "OR"?
2. How does negation affect the truth value of a statement?
3. Can we simplify \( p \rightarrow (q \lor \sim p) \) using logical equivalences?
4. How do truth tables help in understanding logical arguments?
5. What are other common truth tables used in propositional logic?

#### Tip:
Remember that for a conditional statement \( p \rightarrow q \), if \( p \) is false, the statement is always true regardless of \( q \).

Construct a truth table for p → (q ∨ ~p). Use T for true and F for false.

Learn how to construct a truth table for the expression p → (q ∨ ~p). This step-by-step solution explores key logical operations such as negation, disjunction, and implication, essential in understanding propositional logic.

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