Question

The negation of $p \wedge (q \rightarrow r)$ is

• A. $\sim p \wedge (\sim q \rightarrow \sim r)$
• B. $\sim p \vee (q \wedge \sim r)$
• C. $\sim p \vee (\sim q \rightarrow \sim r)$
• D. $p \vee (\sim p \vee r)$

Hint:

To remember this shortcut quickly: the negation of an implication is always "the first stays the same AND the second is negated" ($\sim(q \rightarrow r) \equiv q \wedge \sim r$). Combining this with De Morgan's law for the first part ($\sim(p \wedge \dots) \equiv \sim p \vee \dots$) allows you to write down the answer directly without expanding intermediate steps!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks us to find the logical negation of the compound statement $p \wedge (q \rightarrow r)$ and simplify it using logical equivalence laws.

Step 2: Key Formula or Approach:
We will use the following standard logical equivalence rules:

• Conditional Law: $q \rightarrow r \equiv \sim q \vee r$

• De Morgan's Laws: $$\sim (A \wedge B) \equiv \sim A \vee \sim B$$ $$\sim (A \vee B) \equiv \sim A \wedge \sim B$$

• Double Negation Law: $\sim (\sim q) \equiv q$

Step 3: Detailed Explanation:
Let's write down the full negation expression and apply the rules step-by-step: $$\text{Negation} = \sim [p \wedge (q \rightarrow r)]$$ First, replace the internal conditional implication arrow using the conditional law: $$\equiv \sim [p \wedge (\sim q \vee r)]$$ Now, apply De Morgan's Law across the main conjunction ($\wedge$) operator: $$\equiv (\sim p) \vee \sim (\sim q \vee r)$$ Next, apply De Morgan's Law inside the second bracket to distribute the negation across the disjunction ($\vee$): $$\equiv \sim p \vee (\sim(\sim q) \wedge \sim r)$$ Simplify the double negation component $\sim(\sim q)$ to $q$: $$\equiv \sim p \vee (q \wedge \sim r)$$ This matches the logical statement in option (B).

Step 4: Final Answer:
The negation of the given statement is $\sim p \vee (q \wedge \sim r)$, which corresponds to option (B).