For any statement,p and q, prove that pimplies q -= `(~p vv q)`.

Answer

To prove that \( p \implies q \) is logically equivalent to \( \neg p \lor q \), we will use a truth table. Here’s a step-by-step solution:

### Step 1: Understand the expressions
We need to prove that the implication \( p \implies q \) is equivalent to the disjunction \( \neg p \lor q \).

### Step 2: Construct the truth table
We will create a truth table that includes all possible truth values for the statements \( p \) and \( q \).

| \( p \) | \( q \) | \( \neg p \) | \( p \implies q \) | \( \neg p \lor q \) |
|---------|---------|---------------|---------------------|----------------------|
| T | T | F | T | T |
| T | F | F | F | F |
| F | T | T | T | T |
| F | F | T | T | T |

### Step 3: Fill in the truth values
1. **Column for \( p \) and \( q \)**: We list all combinations of truth values for \( p \) and \( q \).
2. **Column for \( \neg p \)**: This column shows the negation of \( p \).
3. **Column for \( p \implies q \)**: This column is filled based on the definition of implication:
- \( p \implies q \) is false only when \( p \) is true and \( q \) is false (T, F).
- In all other cases, it is true.
4. **Column for \( \neg p \lor q \)**: This column shows the result of the disjunction:
- It is true if at least one of \( \neg p \) or \( q \) is true.

### Step 4: Compare the columns
Now we compare the columns for \( p \implies q \) and \( \neg p \lor q \):

- For \( (T, T) \): Both are T
- For \( (T, F) \): Both are F
- For \( (F, T) \): Both are T
- For \( (F, F) \): Both are T

### Step 5: Conclusion
Since the columns for \( p \implies q \) and \( \neg p \lor q \) have the same truth values for all combinations of \( p \) and \( q \), we conclude that:

\[ p \implies q \equiv \neg p \lor q \]