48 lines
1 KiB
Markdown
48 lines
1 KiB
Markdown
---
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categories:
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- Mathematics
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tags: [logic]
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---
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> Two sentences, P and Q, are truth-functionally equivalent if and only if there is no truth assignment in which P is true and Q is false
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### Informal expression
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```
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P: If it is raining then the pavement will be wet.
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Q: The pavement is not wet unless it is raining.
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```
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### Formal expression
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$$
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P \supset Q \equiv \sim P \lor Q
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$$
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### Truth-tables
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```
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P Q P ⊃ Q
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T T T
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T F F
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F T T
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F F T
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```
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```
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P Q ~ P ∨ Q
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T T T
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T F F
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F T T
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F F T
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```
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### Derivation
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> Propositions $P$ and $Q$ are equivalent in a system of [derivation](Formal%20proofs%20in%20propositional%20logic.md) for propositional logic if $Q$ is derivable from $P$ and $P$ is derivable from $Q$.
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Note that the property of equivalence stated in terms of derivablity above is identical to the derivation rule for the [material biconditional](Biconditional%20Introduction.md):
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//TODO: Add demonstration of this by deriving two equivalents from one of DeMorgan's Laws
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