Question

For three simple statements p, q, and r, $p \rightarrow (q \lor r)$ is logically equivalent to

• A. $(p \lor q) \rightarrow r$
• B. $(p \rightarrow \sim q) \land (p \rightarrow r)$
• C. $(p \rightarrow q) \lor (p \rightarrow r)$
• D. $(p \rightarrow q) \land (p \rightarrow \sim r)$

Answer 1

Answer: (C)

(C)

Answer 2

To find the logical equivalent of the statement $p \rightarrow (q \lor r)$, we can transform the statement using standard logical equivalences. The key equivalence is $A \rightarrow B \equiv \sim A \lor B$.

Applying this equivalence to the given statement: $p \rightarrow (q \lor r) \equiv \sim p \lor (q \lor r)$ Using the associative property of disjunction ($\lor$), we can write this as: $\equiv \sim p \lor q \lor r$

Now, we examine each option and transform it into a similar form using $\sim$ and $\lor$ to check for equivalence.

(A) $(p \lor q) \rightarrow r$ Using $A \rightarrow B \equiv \sim A \lor B$: $(p \lor q) \rightarrow r \equiv \sim (p \lor q) \lor r$ Using De Morgan's Law, $\sim (A \lor B) \equiv \sim A \land \sim B$: $\equiv (\sim p \land \sim q) \lor r$ This is not equivalent to $\sim p \lor q \lor r$.

(B) $(p \rightarrow \sim q) \land (p \rightarrow r)$ Using $A \rightarrow B \equiv \sim A \lor B$ for both implications: $(p \rightarrow \sim q) \land (p \rightarrow r) \equiv (\sim p \lor \sim q) \land (\sim p \lor r)$ Using the distributive property $(A \lor B) \land (A \lor C) \equiv A \lor (B \land C)$: $\equiv \sim p \lor (\sim q \land r)$ This is not equivalent to $\sim p \lor q \lor r$.

(C) $(p \rightarrow q) \lor (p \rightarrow r)$ Using $A \rightarrow B \equiv \sim A \lor B$ for both implications: $(p \rightarrow q) \lor (p \rightarrow r) \equiv (\sim p \lor q) \lor (\sim p \lor r)$ Using the associative and commutative properties of disjunction: $\equiv \sim p \lor q \lor \sim p \lor r$ $\equiv (\sim p \lor \sim p) \lor q \lor r$ Using the idempotent property $A \lor A \equiv A$: $\equiv \sim p \lor q \lor r$ This is equivalent to the original statement.

(D) $(p \rightarrow q) \land (p \rightarrow \sim r)$ Using $A \rightarrow B \equiv \sim A \lor B$ for both implications: $(p \rightarrow q) \land (p \rightarrow \sim r) \equiv (\sim p \lor q) \land (\sim p \lor \sim r)$ Using the distributive property $(A \lor B) \land (A \lor C) \equiv A \lor (B \land C)$: $\equiv \sim p \lor (q \land \sim r)$ This is not equivalent to $\sim p \lor q \lor r$.

Comparing the simplified forms, only option (C) is logically equivalent to the original statement.