Question

The statement pattern \((p∨q)\rightarrow \sim r\) is logically equivalent to

• A. \((\sim p∨\sim q)∨\sim r\)
• B. \((\sim p∧\sim q)∧\sim r\)
• C. \((\sim p∧\sim q)∨\sim r\)
• D. \((\sim p∨\sim q)∧\sim r\)

Hint:

Use a implies b = not a or b, then apply De Morgan's law to not (p or q).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need a statement pattern that is logically equivalent to \((p \lor q) \to \sim r\).

Step 2: Key Formula or Approach:
An implication can be rewritten as \(a \to b \equiv \sim a \lor b\). De Morgan's law says \(\sim(p \lor q) \equiv \sim p \land \sim q\).

Step 3: Apply the implication rule:
Take \(a = p \lor q\) and \(b = \sim r\): \[ (p \lor q) \to \sim r \equiv \sim(p \lor q) \lor \sim r \]

Step 4: Apply De Morgan's law:
\[ \sim(p \lor q) \lor \sim r \equiv (\sim p \land \sim q) \lor \sim r \] This is option (C).

Step 5: Check the other options:
Take \(p=T, q=F, r=T\). The original gives \(T \to F = F\), but option (A) gives \(F \lor T \lor F = T\), so (A) differs. Now take \(p=F, q=F, r=T\). The original gives \(F \to F = T\). Option (B) gives \(T \land T \land F = F\), and option (D) gives \((T)\land F = F\). So (B) and (D) also differ.

Final Answer:
The pattern is equivalent to \((\sim p \land \sim q) \lor \sim r\), option (C). \[ \boxed{(\sim p\land\sim q)\lor\sim r} \]