Question

The negation of the inverse of $\sim p \rightarrow q$ is

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

Hint:

Save time by writing down the conditional definition directly: $A \rightarrow B$ is equivalent to $\sim A \vee B$. The inverse of $\sim p \rightarrow q$ is $p \rightarrow \sim q$, which means $\sim p \vee \sim q$. Taking the negation of $(\sim p \vee \sim q)$ using De Morgan's Law flips everything directly to $p \wedge q$!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given a logical implication statement $\sim p \rightarrow q$. We need to write its inverse statement, and then evaluate the logical negation of that inverse.

Step 2: Key Formula or Approach:
Let's use the rules of formal mathematical logic transformations:
The

Inverse of a conditional statement $A \rightarrow B$ is found by negating both individual parts: $\sim A \rightarrow \sim B$.
The

Negation of any conditional statement $X \rightarrow Y$ is logically equivalent to: $\sim(X \rightarrow Y) \equiv X \wedge \sim Y$.

Step 3: Detailed Explanation:
Let's apply these logical definitions step-by-step:
1. Find the inverse of the statement $\sim p \rightarrow q$:
$$\text{Inverse} = \sim(\sim p) \rightarrow \sim q$$ Using the law of double negation, $\sim(\sim p) \equiv p$, so the inverse statement simplifies to:
$$\text{Inverse} = p \rightarrow \sim q$$ 2. Now, take the negation of this inverse statement:
$$\text{Negation of Inverse} = \sim(p \rightarrow \sim q)$$ Using the conditional equivalence property $\sim(A \rightarrow B) \equiv A \wedge \sim B$:
$$\sim(p \rightarrow \sim q) \equiv p \wedge \sim(\sim q)$$ Simplify the double negation of $q$:
$$p \wedge q$$ This matches option (D).

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