Question

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

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

Hint:

Logic Tip: The phrase "A implies B" means "If A happens, B must happen." The ONLY way to prove this false (negate it) is if A happens AND B does not happen. Therefore, $\sim(A \rightarrow B) = A \text{ AND } (\text{NOT } B)$.

Answer 1

The Correct Option is B

Concept:
To find the negation of a logical implication, we use the logical equivalence $\sim(A \rightarrow B) \equiv A \wedge \sim B$. This is because an implication is only false when the premise ($A$) is true and the conclusion ($B$) is false. De Morgan's laws are then used to distribute the negation over disjunctions ($\vee$) or conjunctions ($\wedge$).
Step 1: Apply the negation rule for implication.
Let the given statement be $S \equiv (p\wedge q)\rightarrow(\sim p\vee r)$. We need to find $\sim S$: $$\sim S \equiv \sim\left( (p\wedge q) \rightarrow (\sim p\vee r) \right)$$ Using $\sim(A \rightarrow B) \equiv A \wedge \sim B$, where $A = (p \wedge q)$ and $B = (\sim p \vee r)$: $$\sim S \equiv (p \wedge q) \wedge \sim(\sim p \vee r)$$
Step 2: Apply De Morgan's Law to the second part.
De Morgan's law states that $\sim(X \vee Y) \equiv \sim X \wedge \sim Y$. Apply this to $\sim(\sim p \vee r)$: $$\sim(\sim p \vee r) \equiv \sim(\sim p) \wedge \sim r$$ The double negation $\sim(\sim p)$ simplifies to just $p$: $$\sim(\sim p \vee r) \equiv p \wedge \sim r$$
Step 3: Combine and simplify using Idempotent Law.
Substitute the simplified second part back into the expression: $$\sim S \equiv (p \wedge q) \wedge (p \wedge \sim r)$$ Because the conjunction operator ($\wedge$) is associative and commutative, we can rearrange the terms: $$\sim S \equiv (p \wedge p) \wedge q \wedge \sim r$$ By the Idempotent Law, $p \wedge p \equiv p$: $$\sim S \equiv p \wedge q \wedge \sim r$$