From b0e8f974e3b3d71a946e227c9e4228472dba296c Mon Sep 17 00:00:00 2001 From: eggy Date: Fri, 13 Jan 2023 11:21:04 -0500 Subject: [PATCH] ece124: add postulates --- docs/1b/ece124.md | 23 +++++++++++++++++++++++ 1 file changed, 23 insertions(+) diff --git a/docs/1b/ece124.md b/docs/1b/ece124.md index dd295fe..f9e28e8 100644 --- a/docs/1b/ece124.md +++ b/docs/1b/ece124.md @@ -45,3 +45,26 @@ The **NOT** operator returns the opposite of its singular input. $$\overline A \text{ or } A'$$ (Source: Wikimedia Commons) + +### Postulates + +In binary algebra, if $x,y,z\in\mathbb B$ such that $\mathbb B=\{0, 1\}$: + +The **identity element** for **AND** $1$ is such that any $x\cdot 1 = x$. + +The **identity element** for **OR** $0$ is such that any $x + 0 = x$. + +In this space, it can be deduced that $x+x'=1$ and $x\cdot x'=0$. + +**De Morgan's laws** are much easier to express in boolean algebra, and denote distributing a negation by flipping the operator: + +$$ +(x\cdot y)'=x'+y' \\ +(x+y)=x'\cdot y' +$$ + +Please see [ECE 108: Discrete Math 1#Operator laws](/1b/ece108/#operator-laws) for more information. + +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.