To determine the logical equivalence of the expression \( \neg p \lor q \), we can use a truth table to analyze the values of \( p \) and \( q \) and their implications.
### Step-by-Step Solution:
1. **Define the Variables**:
Let \( p \) and \( q \) be two logical statements. The possible values for each variable are True (T) or False (F).
2. **Construct the Truth Table**:
We will create a truth table that includes the columns for \( p \), \( q \), \( \neg p \), \( \neg p \lor q \), \( p \implies q \), \( q \implies p \), \( \neg p \implies q \), and \( \neg q \implies p \).
| \( p \) | \( q \) | \( \neg p \) | \( \neg p \lor q \) | \( p \implies q \) | \( q \implies p \) | \( \neg p \implies q \) | \( \neg q \implies p \) |
|---------|---------|---------------|----------------------|---------------------|---------------------|--------------------------|--------------------------|
| T | T | F | T | T | T | T | T |
| T | F | F | F | F | T | T | T |
| F | T | T | T | T | F | T | F |
| F | F | T | T | T | T | F | T |
3. **Evaluate Each Column**:
- **Negation of \( p \)**: This column is True when \( p \) is False and vice versa.
- **Disjunction \( \neg p \lor q \)**: This column is True if either \( \neg p \) is True or \( q \) is True.
- **Implication \( p \implies q \)**: This is False only when \( p \) is True and \( q \) is False.
- **Implication \( q \implies p \)**: This is False only when \( q \) is True and \( p \) is False.
- **Implication \( \neg p \implies q \)**: This is False only when \( \neg p \) is True and \( q \) is False.
- **Implication \( \neg q \implies p \)**: This is False only when \( \neg q \) is True and \( p \) is False.
4. **Compare the Columns**:
Now we need to compare the column for \( \neg p \lor q \) with the columns for the implications:
- The column for \( \neg p \lor q \) is: T, F, T, T
- The column for \( p \implies q \) is: T, F, T, T
- The column for \( q \implies p \) is: T, T, F, T
- The column for \( \neg p \implies q \) is: T, T, T, F
- The column for \( \neg q \implies p \) is: T, T, F, T
From our comparison, we see that \( \neg p \lor q \) is logically equivalent to \( p \implies q \).
### Conclusion:
Thus, the expression \( \neg p \lor q \) is logically equal to \( p \implies q \).
