construct truth table for `(p ^^q) v (~p ^^ q)`

Answer

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

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

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

### Step 3: Create the Table Structure
We will create a table with the following columns:
1. \(p\)
2. \(\neg p\)
3. \(q\)
4. \(R = p \land q\)
5. \(S = \neg p \land q\)
6. \(T = R \lor S\)

### Step 4: Fill in the Values for \(p\) and \(q\)
We will fill in the values for \(p\) and \(q\) in the truth table. The possible combinations are:
- \(p = T\), \(q = T\)
- \(p = T\), \(q = F\)
- \(p = F\), \(q = T\)
- \(p = F\), \(q = F\)

### Step 5: Calculate \(\neg p\)
We will compute the negation of \(p\) for each row:
- If \(p = T\), then \(\neg p = F\)
- If \(p = F\), then \(\neg p = T\)

### Step 6: Calculate \(R = p \land q\)
We will compute \(R\) (the conjunction of \(p\) and \(q\)):
- \(R = T\) if both \(p\) and \(q\) are true, otherwise \(R = F\).

### Step 7: Calculate \(S = \neg p \land q\)
We will compute \(S\) (the conjunction of \(\neg p\) and \(q\)):
- \(S = T\) if both \(\neg p\) and \(q\) are true, otherwise \(S = F\).

### Step 8: Calculate \(T = R \lor S\)
Finally, we will compute \(T\) (the disjunction of \(R\) and \(S\)):
- \(T = T\) if either \(R\) or \(S\) is true, otherwise \(T = F\).

### Step 9: Complete the Truth Table
Now we can fill in the truth table with the calculated values.

| \(p\) | \(\neg p\) | \(q\) | \(R = p \land q\) | \(S = \neg p \land q\) | \(T = R \lor S\) |
|-------|------------|-------|--------------------|------------------------|-------------------|
| T | F | T | T | F | T |
| T | F | F | F | F | F |
| F | T | T | F | T | T |
| F | T | F | F | F | F |

### Final Truth Table
The final truth table for the expression \((p \land q) \lor (\neg p \land q)\) is:

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