From 78b8fbc2c9d3a3672e94bd2361e68d6e4e031168 Mon Sep 17 00:00:00 2001 From: eggy Date: Tue, 17 Jan 2023 22:22:45 -0500 Subject: [PATCH] ece124: add base logic --- docs/1b/ece124.md | 25 +++++++++++++++++++++++++ 1 file changed, 25 insertions(+) diff --git a/docs/1b/ece124.md b/docs/1b/ece124.md index bc59fe7..6b1c363 100644 --- a/docs/1b/ece124.md +++ b/docs/1b/ece124.md @@ -65,6 +65,31 @@ $$ Please see [ECE 108: Discrete Math 1#Operator laws](/1b/ece108/#operator-laws) for more information. +AND and OR are commutative. + +- $x\cdot y=y\cdot x$ +- $x+y=y+x$ + +AND and OR are associative. + +- $x\cdot(y\cdot z)=(x\cdot y)\cdot z)$ +- ... + +AND and OR are distributive with each other. + +- $x\cdot (y+z)=x\cdot y+z\cdot z$ + +A term that depends on another term ORed together can be "absorbed". + +- $x+x\cdot y=x$ +- $x\cdot(x+y)=x$ + +If a term being true also results in other ORed terms being true, it is redundant and can be eliminated via consensus. + +- $x\cdot y+y\cdot z+x'\cdot z=x\cdot y+x'\cdot z$ + - if y and z are true, at least one of the other two terms must be true +- $(x+y)\cdot (y+z)\cdot(x'+z)=(x+y)\cdot (x'+z)$ + The **synthesis** of an algebraic formula represents its implementation via logic gates. In this course, its total cost is the sum of all inputs to all gates and the number of gates, *excluding* initial inputs of "true" or an initial negation. In order to deduce an algebraic expression from a truth table, **OR** all of the rows in which the function returns true and simplify.