`(p^^~q) ^^ (~pvvq)` is

Options

A. a contradiction
B. a tautology
C. either (1) or (2)
D. neither (1) nor (2)

Explanation

To solve the expression `(p ∧ ¬q) ∧ (¬p ∨ q)`, we will construct a truth table to evaluate the expression step by step.

### Step 1: Identify the variables
We have two variables: `p` and `q`.

### Step 2: List all possible truth values
Since there are two variables, the possible combinations of truth values for `p` and `q` are:
1. `p = True`, `q = True`
2. `p = True`, `q = False`
3. `p = False`, `q = True`
4. `p = False`, `q = False`

### Step 3: Calculate ¬q and ¬p
We need to find the negations of `q` and `p` for each combination:
- If `q = True`, then `¬q = False`
- If `q = False`, then `¬q = True`
- If `p = True`, then `¬p = False`
- If `p = False`, then `¬p = True`

### Step 4: Create the truth table
We will now create a truth table to evaluate the expression `(p ∧ ¬q) ∧ (¬p ∨ q)`.

| p | q | ¬p | ¬q | p ∧ ¬q | ¬p ∨ q | (p ∧ ¬q) ∧ (¬p ∨ q) |
|-------|-------|-------|-------|--------|--------|---------------------|
| True | True | False | False | False | True | False |
| True | False | False | True | True | False | False |
| False | True | True | False | False | True | False |
| False | False | True | True | False | True | False |

### Step 5: Analyze the results
From the truth table, we can see that the final column `(p ∧ ¬q) ∧ (¬p ∨ q)` has all values as `False`. This means that the expression is always false, which indicates that it is a contradiction.

### Conclusion
The expression `(p ∧ ¬q) ∧ (¬p ∨ q)` is a contradiction.