43 lines
1.4 KiB
Markdown
43 lines
1.4 KiB
Markdown
---
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categories:
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- Logic
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tags: []
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---
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# DeMorgan's Laws
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DeMorgan's laws express some fundamental equivalences that obtain between the
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Boolean
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[connectives](/Logic/Propositional_logic/Truth-functional_connectives.md).
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## First Law
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> The negation of a conjunction is logically equivalent to the disjunction of
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> the negations of the original conjuncts.
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$$
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\lnot (P \land Q) \leftrightarrow \lnot P \lor \lnot Q
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$$
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The equivalence is demonstrated with the following truth-table
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| $P$ | $Q$ | $ \lnot (P \land Q)$ | $ \lnot P \lor \lnot Q$ |
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| --- | --- | -------------------- | ----------------------- |
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| T | T | F | F |
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| T | F | T | T |
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| F | T | T | T |
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| F | F | T | T ### Truth conditions |
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> The negation of a disjunction is equivalent to the conjunction of the negation
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> of the original disjuncts.
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$$
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\lnot (P \lor Q) \leftrightarrow \lnot P \land \lnot Q
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$$
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| $P$ | $Q$ | $ \lnot (P \lor Q)$ | $ \lnot P \land \lnot Q$ |
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| --- | --- | ------------------- | ------------------------ |
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| T | T | F | F |
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| T | F | F | F |
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| F | T | F | F |
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| F | F | T | T |
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