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ece124: add postulates

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eggy 2023-01-13 11:21:04 -05:00
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@ -45,3 +45,26 @@ The **NOT** operator returns the opposite of its singular input.
$$\overline A \text{ or } A'$$
<img src="https://upload.wikimedia.org/wikipedia/commons/6/60/NOT_ANSI_Labelled.svg" width=200>(Source: Wikimedia Commons)</img>
### 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.