Logical Equivalence
Logical Equivalence: If two compound statements have the same truth values for all combinations of their component statements, then we say they are logically equivalent/
The text uses the symbol \(\equiv\), but we'll use \(\iff\) in this course.
Theorem 2.17
If \(P, Q\) aare statements (or open sentences), then
\[ \begin{gather} P \implies Q\\ \iff\\ (\neg{P}) \vee Q \end{gather} \]
Proof:
| \(P\) | \(Q\) | \(P \implies Q\) | \((\neg{P}) \vee Q\) |
|---|---|---|---|
| T | T | T | T |
| T | F | F | F |
| F | T | T | T |
| F | F | T | T |