Question

If the truth value of the statement pattern \((\sim p∧q)∨(\sim p∧\sim q)∨(p∧\sim q)\) is \(F\), then the truth values of \((p∨\sim q)\) and \((p\rightarrow q)\) are \(\ldots\) respectively.

• A. \(F,F\)
• B. \(F,T\)
• C. \(T,T\)
• D. \(T,F\)

Hint:

Simplify the pattern: it equals ~(p and q). It is false only when p and q are both true.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We simplify the compound statement and use the fact that it is false to find the truth values of \(p\) and \(q\).

Step 2: Key Formula or Approach:
Combine the terms with common factors using the distributive law.

Step 3: Detailed Explanation:
\[ (\sim p \wedge q) \vee (\sim p \wedge \sim q) = \sim p \wedge (q \vee \sim q) = \sim p \wedge T = \sim p \]
So the pattern becomes \(\sim p \vee (p \wedge \sim q)\).
\[ \sim p \vee (p \wedge \sim q) = (\sim p \vee p) \wedge (\sim p \vee \sim q) = T \wedge (\sim p \vee \sim q) = \sim p \vee \sim q = \sim(p \wedge q) \]
The pattern is false only when \(p \wedge q\) is true, that is \(p = T\) and \(q = T\).
Now evaluate:
\(p \vee \sim q = T \vee F = T\).
\(p \rightarrow q = T \rightarrow T = T\).
So the two truth values are T and T. Options (A), (B) and (D) each contain an F, which cannot happen here.

Final Answer:
The truth values are T, T, option (C). \[ \boxed{T,T \text{ (C)}} \]