Prove that \( p \Leftrightarrow q = q \Leftrightarrow p \).
To prove this equivalence, a truth table is constructed to evaluate the truth values for both \( p \Leftrightarrow q \) and \( q \Leftrightarrow p \) across all possible combinations of \( p \) and \( q \).
Truth Table for Bi-conditional Equivalence:
| \( p \) | \( q \) | \( p \Leftrightarrow q \) | \( q \Leftrightarrow p \) |
|---|---|---|---|
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 |
Conclusion:
From the truth table above, it is found that both propositions \( p \Leftrightarrow q \) and \( q \Leftrightarrow p \) possess the same truth set: (1, 0, 0, 1).
Hence, it is proved that: \( p \Leftrightarrow q = q \Leftrightarrow p \).
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