Question

Which of the following logical statements is a tautology?

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

Hint:

A tautology is true for every combination of truth values; test modus tollens.

Answer 1

The Correct Option is A

Step 1: Truth Table for (A):
Row TT: $p\rightarrow q=T$, $\sim q=F$, antecedent F, so whole is T. Row TF: $p\rightarrow q=F$, antecedent F, whole T. Row FT: $\sim q=F$, antecedent F, whole T. Row FF: antecedent $T\wedge T=T$, $\sim p=T$, whole T.

Step 2: Conclusion for (A):
All four rows give T, so (A) is a tautology.

Step 3: Rule Out the Rest:
(B) and (C) are always false. (D) contains the factor $r$, so it is false for $r=F$. Option (A).

Final Answer:
Option (A). \[ \boxed{\text{(A) } [(p\rightarrow q)\wedge\sim q]\rightarrow\sim p} \]