Construct truth tables for each of the following :
`qimplies [(~p)vv q]`

Answer

To construct the truth table for the expression \( q \implies [(\sim p) \lor q] \), we will follow these steps:

### Step 1: Identify Variables
We have two variables, \( p \) and \( q \).

### Step 2: Determine the Number of Rows
Since we have 2 variables, the number of rows in the truth table will be \( 2^n = 2^2 = 4 \).

### Step 3: Create the Truth Table Structure
We will create a table with the following columns:
1. \( p \)
2. \( q \)
3. \( \sim p \) (negation of \( p \))
4. \( (\sim p) \lor q \) (disjunction of \( \sim p \) and \( q \))
5. \( q \implies [(\sim p) \lor q] \)

### Step 4: Fill in the Values for \( p \) and \( q \)
We will fill in the values for \( p \) and \( q \) in the truth table:

| \( p \) | \( q \) |
|---------|---------|
| T | T |
| T | F |
| F | T |
| F | F |

### Step 5: Calculate \( \sim p \)
Now, we will calculate \( \sim p \):

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

### Step 6: Calculate \( (\sim p) \lor q \)
Next, we will calculate \( (\sim p) \lor q \):

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

### Step 7: Calculate \( q \implies [(\sim p) \lor q] \)
Finally, we will calculate \( q \implies [(\sim p) \lor q] \). The implication \( A \implies B \) is false only when \( A \) is true and \( B \) is false; otherwise, it is true.

| \( p \) | \( q \) | \( \sim p \) | \( (\sim p) \lor q \) | \( q \implies [(\sim p) \lor q] \) |
|---------|---------|---------------|------------------------|-------------------------------------|
| T | T | F | T | T |
| T | F | F | F | T |
| F | T | T | T | T |
| F | F | T | T | T |

### Final Truth Table
The complete truth table is:

| \( p \) | \( q \) | \( \sim p \) | \( (\sim p) \lor q \) | \( q \implies [(\sim p) \lor q] \) |
|---------|---------|---------------|------------------------|-------------------------------------|
| T | T | F | T | T |
| T | F | F | F | T |
| F | T | T | T | T |
| F | F | T | T | T |