Question

Consider the two statements :
(S1) : \((p \to q) \lor (\neg q \to p)\) is a tautology.
(S2) : \((p \land \neg q) \land (\neg p \lor q)\) is a fallacy.
Then :

• A. only (S1) is true.
• B. only (S2) is true.
• C. both (S1) and (S2) are false.
• D. both (S1) and (S2) are true.

Hint:

Recognizing standard logical forms can save a lot of time over constructing full truth tables. Be familiar with equivalences like \(p \to q \equiv \neg p \lor q\) and \( \neg (p \land \neg q) \). Spotting patterns like \(A \land \neg A\) (fallacy) or \(A \lor \neg A\) (tautology) is a key skill.

Answer 1

The Correct Option is D

Step 1: Understanding the Question
We need to determine the logical nature of two compound statements, (S1) and (S2). A statement is a tautology if it is always true, and a fallacy (or contradiction) if it is always false.
Step 2: Key Formula or Approach
We will use the laws of logical equivalence to simplify each statement. Key equivalences: \(A \to B \equiv \neg A \lor B\) De Morgan's Laws: \(\neg(A \land B) \equiv \neg A \lor \neg B\) and \(\neg(A \lor B) \equiv \neg A \land \neg B\)
Step 3: Detailed Explanation
Analyze Statement (S1):
(S1): \((p \to q) \lor (\neg q \to p)\) Using the equivalence \(A \to B \equiv \neg A \lor B\): \[ (p \to q) \equiv \neg p \lor q \] \[ (\neg q \to p) \equiv \neg(\neg q) \lor p \equiv q \lor p \] So, (S1) becomes: \[ (\neg p \lor q) \lor (q \lor p) \] Using commutative and associative laws for \(\lor\): \[ (\neg p \lor p) \lor (q \lor q) \] We know that \(\neg p \lor p\) is a tautology (T) and \(q \lor q \equiv q\). \[ T \lor q \] Since T OR anything is T, the statement simplifies to T. Thus, (S1) is a tautology. The statement "(S1) is a tautology" is true.
Analyze Statement (S2):
(S2): \((p \land \neg q) \land (\neg p \lor q)\) Let's analyze the second part of the conjunction: \((\neg p \lor q)\). This is equivalent to \((p \to q)\). Also, note that \(\neg(p \land \neg q) \equiv \neg p \lor \neg(\neg q) \equiv \neg p \lor q\). So, let \(A = p \land \neg q\). Then \((\neg p \lor q) \equiv \neg A\). The statement (S2) has the form: \[ A \land \neg A \] This is the definition of a contradiction, which is always false. Thus, (S2) is a fallacy. The statement "(S2) is a fallacy" is true.
Step 4: Final Answer
Both statements, that (S1) is a tautology and (S2) is a fallacy, are true statements.