The proposition `(p to ~p) ^^ (~p to p)` is a

Options

A. a tautology
B. a contradiction
C. neither a tauology nor a contradiction
D. a tautology and a contradition

Explanation

To determine the truth value of the proposition `(p → ~p) ∧ (~p → p)`, we will analyze each part step by step.

### Step 1: Rewrite the implications
The implication \( p \to \neg p \) can be rewritten using the logical equivalence \( p \to q \equiv \neg p \lor q \).
Thus, we have:
\[ p \to \neg p \equiv \neg p \lor \neg p \equiv \neg p \]
Similarly, for \( \neg p \to p \):
\[ \neg p \to p \equiv \neg(\neg p) \lor p \equiv p \lor p \equiv p \]

### Step 2: Substitute back into the proposition
Now we can substitute these results back into the original proposition:
\[ (p \to \neg p) \land (\neg p \to p) \equiv \neg p \land p \]

### Step 3: Analyze the conjunction
The expression \( \neg p \land p \) represents a contradiction because \( p \) cannot be both true and false at the same time. Therefore, this expression is always false.

### Conclusion
The proposition `(p → ~p) ∧ (~p → p)` is a contradiction, which means it is always false.

### Summary of Steps
1. Rewrite the implications using logical equivalences.
2. Substitute the rewritten implications back into the original proposition.
3. Analyze the resulting conjunction to determine its truth value.