Let p and q be any two propositons
Statement 1 : `(p rarr q) harr q vv ~p` is a tautology
Statement 2 : `~(~p ^^ q) ^^ (p vv q) harr p` is fallacy

Options

A. 1) Both statement 1 and statement 2 are true
B. 2) Both statement 1 and statement 2 are false
C. 3) statement 1 is true and statement 2 is false
D. 4) statement 1 is false and statement 2 is true

Answer

To determine whether the given statements are tautologies or fallacies, we will construct truth tables for each statement.

### Statement 1: \((p \rightarrow q) \leftrightarrow (q \lor \neg p)\)

1. **Construct the truth table:**
- List all possible truth values for \(p\) and \(q\):
- \(p\): T, T, F, F
- \(q\): T, F, T, F
- Calculate \(p \rightarrow q\):
- \(p \rightarrow q\) is false only when \(p\) is true and \(q\) is false.
- Values: T, F, T, T
- Calculate \(\neg p\):
- Negation of \(p\): F, F, T, T
- Calculate \(q \lor \neg p\):
- This is true if either \(q\) or \(\neg p\) is true.
- Values: T, F, T, T
- Calculate \((p \rightarrow q) \leftrightarrow (q \lor \neg p)\):
- This is true if both sides have the same truth value.
- Values: T, F, T, T

2. **Final column of Statement 1:**
- The final truth values are T, F, T, T.
- Since there are false values, Statement 1 is **not a tautology**.

### Statement 2: \(\neg(\neg p \land q) \land (p \lor q) \leftrightarrow p\)

1. **Construct the truth table:**
- List all possible truth values for \(p\) and \(q\):
- \(p\): T, T, F, F
- \(q\): T, F, T, F
- Calculate \(\neg p\):
- Negation of \(p\): F, F, T, T
- Calculate \(\neg p \land q\):
- This is true only if both \(\neg p\) and \(q\) are true.
- Values: F, F, T, F
- Calculate \(\neg(\neg p \land q)\):
- Negation of the previous result.
- Values: T, T, F, T
- Calculate \(p \lor q\):
- This is true if either \(p\) or \(q\) is true.
- Values: T, T, T, F
- Calculate \(\neg(\neg p \land q) \land (p \lor q)\):
- This is true if both sides are true.
- Values: T, T, F, F
- Calculate \(\neg(\neg p \land q) \land (p \lor q) \leftrightarrow p\):
- This is true if both sides have the same truth value.
- Values: T, T, F, F

2. **Final column of Statement 2:**
- The final truth values are T, T, F, F.
- Since there are false values, Statement 2 is **not a fallacy**.

### Conclusion:
- Statement 1 is **not a tautology**.
- Statement 2 is **not a fallacy**.

### Final Answer:
Both statements are incorrect as per the definitions given in the question.

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