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#logic
## Definition
> [!tldr] Definition
> Two [[propositions]] are said to be **logically equivalent** if they have the same truth value in each variation of truth values in their [[Atomic and molecular propositions|atomic statements]]. If $P$ and $Q$ are logically equivalent, we write $P \equiv Q$.
Notes:
- Two logically equivalent propositions will have the same [[Truth tables|truth tables]] for each possible combination of truth values. In fact, one way to determine if two propositions are logically equivalent, is to construct and then compare their truth tables.
- Variations on the notation $\equiv$ exist. For example, sometimes it's written $P \cong Q$, or $P \sim Q$.
## Examples and Non-Examples
The propositions $P \rightarrow Q$ and $(\neg P) \vee Q$ are logically equivalent, as shown by their truth tables:
| $P$ | $Q$ | $P \rightarrow Q$ |
| -- | -- | ---- |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
| $P$ | $Q$ | $\neg P$ | $(\neg P) \vee Q$ |
| -- | -- | ---- | ---- |
| T | T | F | T
| T | F | F | F
| F | T | T | T
| F | F | T | T
Likewise, one of [[DeMorgan's Laws]] states that $\neg(P \vee Q) \equiv (\neg P) \wedge (\neg Q)$. Here are the truth tables side by side with the final statements indicated with a vertical arrow:
| $P$ | $Q$ | $P \vee Q$ | $\stackrel{\Downarrow}{\neg(P \vee Q)}$ | $\neg P$ | $\neg Q$ | $\stackrel{\Downarrow}{(\neg P) \wedge (\neg Q)}$ |
| ----- | ----- | ---------- | --------------------------------------- | -------- | -------- | ------------------------------------------------- |
| True | True | True | False | False | False | False |
| True | False | True | False | False | True | False |
| False | True | True | False | True | False | False |
| False | False | False | True | True | True | True |
## Resources
- Book section: [Logically equivalent statements](<https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/02%3A_Logical_Reasoning/2.02%3A_Logically_Equivalent_Statements>)
- Tutorial: [Logical equivalence](https://www.csm.ornl.gov/~sheldon/ds/sec1.1.html#:~:text=Two%20statement%20forms%20are%20logically,each%20variation%20of%20statement%20variables.&text=p%20q%20and%20q%20p,so%20they%20are%20logically%20equivalent.)