Maths · Relations, type of relations, equivalence relations
Statement \) is equivalent to Statement \) is a tautology
Statement \( 1: \sim(\mathbf{p} \leftrightarrow \sim \mathbf{q}) \) is equivalent to \( \mathbf{p} \leftrightarrow \mathbf{q} \) Statement \( 2: \sim(\mathbf{p} \leftrightarrow \sim q) \) is a tautology
- A. Both Statement 1 and Statement 2 are true and Statement 2 is a correct explanation for statement 1
- B. Both Statement 1and Statement 2 are true and Statement 2 is not a correct explanation for Statement 1
- C. Statement 1 is true but statement 2 is false
- D. Statement 1 is false but Statement 2 is true
Step-by-step solution
Statement 1: ~(p ↔ ~q) is equivalent to p ↔ q. Using logical equivalence, p ↔ ~q is equivalent to ~(p ↔ q), so negating gives p ↔ q. Thus true. Statement 2: ~(p ↔ ~q) is equivalent to p ↔ q, which is not a tautology because it is false when p and q differ. So false. Hence option C.
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