construct truth table for `(p vv q)vv(r ^^ ~q)`

Answer

To construct the truth table for the expression \((p \lor q) \lor (r \land \neg q)\), we will follow these steps:

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

### Step 2: Determine Number of Rows
Since we have three variables, the number of possible combinations of truth values (True or False) is \(2^3 = 8\). Thus, we will have 8 rows in our truth table.

### Step 3: Create the Table Structure
We will create a table with the following columns:
1. \(p\)
2. \(q\)
3. \(r\)
4. \(\neg q\) (negation of \(q\))
5. \(r \land \neg q\) (conjunction of \(r\) and \(\neg q\))
6. \(p \lor q\) (disjunction of \(p\) and \(q\))
7. \((p \lor q) \lor (r \land \neg q)\) (final expression)

### Step 4: Fill in the Values
We will fill in the truth values for \(p\), \(q\), and \(r\) in all possible combinations:

| \(p\) | \(q\) | \(r\) | \(\neg q\) | \(r \land \neg q\) | \(p \lor q\) | \((p \lor q) \lor (r \land \neg q)\) |
|-------|-------|-------|------------|---------------------|---------------|---------------------------------------|
| T | T | T | F | F | T | T |
| T | T | F | F | F | T | T |
| T | F | T | T | T | T | T |
| T | F | F | T | F | T | T |
| F | T | T | F | F | T | T |
| F | T | F | F | F | T | T |
| F | F | T | T | T | F | T |
| F | F | F | T | F | F | F |

### Step 5: Analyze the Final Column
The final column \((p \lor q) \lor (r \land \neg q)\) shows the truth values of the entire expression for each combination of \(p\), \(q\), and \(r\).

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

| \(p\) | \(q\) | \(r\) | \(\neg q\) | \(r \land \neg q\) | \(p \lor q\) | \((p \lor q) \lor (r \land \neg q)\) |
|-------|-------|-------|------------|---------------------|---------------|---------------------------------------|
| T | T | T | F | F | T | T |
| T | T | F | F | F | T | T |
| T | F | T | T | T | T | T |
| T | F | F | T | F | T | T |
| F | T | T | F | F | T | T |
| F | T | F | F | F | T | T |
| F | F | T | T | T | F | T |
| F | F | F | T | F | F | F |