From 03f25f50a9fc4fbeaab1b6e5aa0bb0174104d17d Mon Sep 17 00:00:00 2001 From: thomasabishop Date: Sun, 18 Dec 2022 19:00:05 +0000 Subject: [PATCH] Autosave: 2022-12-18 19:00:05 --- Logic/Propositional_logic/Boolean_algebra.md | 17 ++++++++++++++++- 1 file changed, 16 insertions(+), 1 deletion(-) diff --git a/Logic/Propositional_logic/Boolean_algebra.md b/Logic/Propositional_logic/Boolean_algebra.md index 681f341..df685fd 100644 --- a/Logic/Propositional_logic/Boolean_algebra.md +++ b/Logic/Propositional_logic/Boolean_algebra.md @@ -91,12 +91,27 @@ $$ \lnot(\lnot(x)) \lor \lnot(\lnot(y)) $$ -Of course now we have two double negatives. We can apply the double negation law to: +Of course now we have two double negatives. We can apply the double negation law to get: $$ x \lor y $$ +### Truth table + +Whenever we simplify an algebraic expression the value of the resulting expression should match that of the complex expression. We can demonstrate this with a truth table: + +| $x$ | $y$ | $\lnot(\lnot(x) \land \lnot (x \lor y))$ | $x \lor y$ | +| --- | --- | ---------------------------------------- | ---------- | +| 0 | 0 | 0 | 0 | +| 0 | 1 | 1 | 1 | +| 1 | 0 | 1 | 1 | +| 1 | 1 | 1 | 1 | + +### Significance for computer architecture + +The fact that we can take a complex Boolean function and reduce it to a simpler formulation has great significance for the development of computer architectures, specifically [logic gates](/Electronics_and_Hardware/Digital_circuits/Logic_gates.md). It would be rather resource intensive and inefficient to create a gate that is representative of the complex function. Whereas the simplified version only requires a single [OR gate](/Electronics_and_Hardware/Digital_circuits/Logic_gates.md#or-gate) + // TO DO: - Use truth tables to show equivalence