Question

The negation of statement pattern $(p \wedge \sim q) \rightarrow (p \vee \sim q)$ is

• A. a tautology
• B. a contingency
• C. a contradiction
• D. equivalent to $p \vee q$

Hint:

$\sim(P \rightarrow Q) \equiv P \wedge \sim Q$. If you test this with a truth table, you will find all entries are False.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to find the logical nature of the negation of a given implication statement.
Step 2: Key Formula or Approach:
The negation of an implication \( X \rightarrow Y \) is \( X \wedge \sim Y \).
Step 3: Detailed Explanation:
Let \( X = (p \wedge \sim q) \) and \( Y = (p \vee \sim q) \).
The negation is \( (p \wedge \sim q) \wedge \sim(p \vee \sim q) \).
Apply De Morgan's Law to \( \sim(p \vee \sim q) \):
\[ \sim(p \vee \sim q) \equiv (\sim p \wedge \sim(\sim q)) \equiv (\sim p \wedge q) \]
Now substitute back into the negation expression:
\[ (p \wedge \sim q) \wedge (\sim p \wedge q) \]
Rearranging using commutative and associative laws:
\[ (p \wedge \sim p) \wedge (q \wedge \sim q) \]
We know \( (p \wedge \sim p) \equiv F \) and \( (q \wedge \sim q) \equiv F \).
\[ F \wedge F \equiv F \]
A statement that is always False is a contradiction.
Step 4: Final Answer:
The negation is a contradiction.