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# Build a truth table for the statement ~$pvq$^~p

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## Answer to a math question Build a truth table for the statement ~$pvq$^~p

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To build a truth table for the given statement ~$pvq$^~p, we need to consider all possible combinations of truth values for p and q.

Step 1: Determine the components of the statement
The given statement consists of three components: ~$pvq$, ~p, and the overall statement.

Step 2: List all possible combinations of truth values for p and q
Since there are two variables, p and q, there are 2^2 = 4 possible combinations of truth values.

| p | q |
|---|---|
| T | T |
| T | F |
| F | T |
| F | F |

Step 3: Evaluate the components
~$pvq$: This component is the negation of the disjunction $pvq$. To evaluate this, we consider the disjunction $pvq$ and then take the negation of the result.

| p | q | pvq | ~$pvq$ |
|---|---|-----|--------|
| T | T | T | F |
| T | F | T | F |
| F | T | T | F |
| F | F | F | T |

~p: This component is the negation of p.

| p | ~p |
|---|----|
| T | F |
| F | T |

Step 4: Calculate the overall statement
The overall statement is the conjunction of ~$pvq$ and ~p.

| p | q | pvq | ~$pvq$ | ~p | $pvq$^~p |
|---|---|-----|--------|----|----------|
| T | T | T | F | F | F |
| T | F | T | F | F | F |
| F | T | T | F | T | F |
| F | F | F | T | T | T |

Step 5: Write the truth table
Finally, we can compile all the information into a truth table for the given statement ~$pvq$^~p.

| p | q | pvq | ~$pvq$ | ~p | $pvq$^~p |
|---|---|-----|--------|----|----------|
| T | T | T | F | F | F |
| T | F | T | F | F | F |
| F | T | T | F | T | F |
| F | F | F | T | T | T |

Answer: The truth table for the statement ~$pvq$^~p is as shown above.

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