:PROPERTIES: :ID: 0f27ed55-e622-428f-83ec-8aa9057d0f9c :mtime: 20231006035204 :ctime: 20231006034730 :END: #+title: Boolean Algebra #+filetags: :public:project: * Equations $a + 0 = a$ $a + a = a$ $a + 1 = 1$ $a + \overline{a} = 0$ * Distributive Laws "and" distributes over "or" $a.(b + c + \cdots) = a.b + a.c + \cdots$ "or" distributes over "and" $a+(b.c.\ldots)= (a+b).(a+c).\ldots$ * Absorbtion Laws $a + a.b = a$ * Consensus Theorem $a.b + \overline{a}.c + b.c = a.b + \overline{a}.c$ $(a + b)$ * Examples of Boolean Simplification Here is an example of simplifying an expression using boolean algebra. $a.(\overline{a} + b) = a.\overline{a} + a.b = 0 + a.b = a.b$ * Proof of the Absorbtion Law TODO * Another Simplifcation Example $x.y+\overline{y}.z + x.z + x.y.z = x.y.z + x.y.\overline{z} + x.y.\overline{z} + x.\overline{y}.z + \overline{x}.\overline{y}.z + x.y.z + x.\overline{y}.z + x.y.z = x.y.z + x.y.\overline{z} + x.\overline{y}.z + \overline{x}.\overline{y}.z = x.y.(z + \overline{z}) + \overline{y}.z.(x + \overline{z}) = x.y.1 + \overline{y}.z.1 = x.y + \overline{y}.z$ * A Proof of the Consensus Theorem $a.b + \overline{a}.c + b.c = a.b + \overline{a}.c + a.b.c$ * DeMorgan's Laws $\overline{a+b+c+\cdots} = \overline{a}.\overline{b}.\overline{c}.\cdots$ and $\overline{a.b.c.\cdots}=\overline{a} + \overline{b} + \overline{c} + \cdots $