From dad4dbd366e57b27e14cd96c4eaee6b9815a22f3 Mon Sep 17 00:00:00 2001 From: thomasabishop Date: Fri, 23 Dec 2022 14:00:05 +0000 Subject: [PATCH] Autosave: 2022-12-23 14:00:05 --- Logic/Laws_and_theorems.md/DeMorgan's_Laws.md | 13 +++++++++---- 1 file changed, 9 insertions(+), 4 deletions(-) diff --git a/Logic/Laws_and_theorems.md/DeMorgan's_Laws.md b/Logic/Laws_and_theorems.md/DeMorgan's_Laws.md index d816931..91635e3 100644 --- a/Logic/Laws_and_theorems.md/DeMorgan's_Laws.md +++ b/Logic/Laws_and_theorems.md/DeMorgan's_Laws.md @@ -6,26 +6,31 @@ tags: [logic, laws] # DeMorgan's Laws -DeMorgan's laws express some fundamental equivalences that obtain between the Boolean [connectives](Truth-functional%20connectives.md): +DeMorgan's laws express some fundamental equivalences that obtain between the Boolean [connectives](/Logic/Propositional_logic/Truth-functional_connectives.md). ## First Law > The negation of a conjunction is logically equivalent to the disjunction of the negations of the original conjuncts. $$ -\sim (P \& Q) \equiv \sim P \lor \sim Q +\lnot (P \land Q) \leftrightarrow \lnot P \lor \lnot Q $$ The equivalence is demonstrated with the following truth-table -![demorgan-1.png](/img/demorgan-1.png) +| $P$ | $Q$ | $ \lnot (P \land Q)$ | $ \lnot P \lor \lnot Q$ | +| --- | --- | -------------------- | ----------------------- | +| T | T | F | F | +| T | F | T | T | +| F | T | T | T | +| F | F | T | T | ## Second Law > The negation of a disjunction is equivalent to the conjunction of the negation of the original disjuncts. $$ -\sim (P \lor Q) \equiv \sim P \& \sim Q +\lnot (P \lor Q) \leftrightarrow \lnot P \land \lnot Q $$ ![demorgan-2.png](/img/demorgan-2.png)