Question

If the truth value of the statement pattern \([p \wedge \sim r] \to \sim r \wedge q\) is False, then which of the following has truth value False?

• A. \((p \vee r) \to \sim r\)
• B. \((r \vee q) \to \sim p\)
• C. \(\sim (p \vee q) \to \sim r\)
• D. \(\sim (r \vee q) \to \sim p\)

Hint:

A conditional statement is false only when its first part is true and its second part is false; use that to fix p, q and r.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
\(A \rightarrow B\) is false only when \(A\) is true and \(B\) is false.

Step 2: Find the Truth Values
For \([p \wedge \sim r] \rightarrow (\sim r \wedge q)\) to be false, \(p \wedge \sim r\) is true, so \(p = T\) and \(r = F\). Also \(\sim r \wedge q\) is false; as \(\sim r = T\), we need \(q = F\).

Step 3: Test the Options
With \(p = T,\ q = F,\ r = F\): (1) \((p \vee r) \rightarrow \sim r\) is \(T \rightarrow T = T\). (2) \((r \vee q) \rightarrow \sim p\) is \(F \rightarrow F = T\). (3) \(\sim(p \vee q) \rightarrow \sim r\) is \(F \rightarrow T = T\). (4) \(\sim(r \vee q) \rightarrow \sim p\) is \(T \rightarrow F = F\).

Step 4: Compare the options
Only the last option is false.

Final Answer:
The last statement, not of (r or q) implies not p, is the false one. \[ \boxed{\text{(D) }\sim (r \vee q) \rightarrow \sim p} \]