Karnaugh Mapping

 There are many laws in Boolean Algebra, but three of the most common  ones are the Associative law, Distributive law, and the Commutative  law. The Associative law states that when there are common operations  then the parenthesis can be move and different variables can be grouped  without issue. This means that if you have A+C+F, you can group  parenthesis about them however you wish and it doesn't matter. You could  group them A+(C+F) or (A+C)+F, the output would still be the same. The  Distributive law allows you to multiply or factor an equation. This  means equations can be simplified from A(C+F)=A.C+A.F or can be taken  from A+(C.F)=(A+C).(A+F). These allow for you to factor out equations  and allow for ease of computations. The commutative law states that  order of application of the variables or terms do not matter when it  comes to the value of the output. This means that when you have multiple  terms with like operations you can change the order without effecting  the output. For example: A+C+F=C+F+A. 

Another law in Boolean algebra is De Morgan's law which helps  simplify mathematical expressions. There are two different parts of this  law that in a way state similar things. The first rule states that when  two or more input variables are connected with an Or and then` negated,  the result is equal to the and of the complements of the input  variables. Here is a truth table for the first law:

The second law states that two or more inputs are combined with an  AND statement and negated then the result will be equal to the OR of the  complements of the individual variables. If that does not make sense,  here is a truth table and diagram to explain: 

For those who like to see the equations written in set theory:

• (A ∪ B)’ = A’ ∩ B’
• (A ∩ B)’ = A’ ∪ B’

These equations can be boiled down to the combination of two inputs  negated are equal to their complement. These equations are helpful as it  allows for more complex expressions to be simplified down with NAND and  NOR gates.

Karnaugh mapping is one way to simplify algebraic expressions without  having to use any sort of theorems or equations manipulations. Its a  visual method that works best with two to four variables. It can be done  with more than 4, but is said to be much more difficult. The Karnaugh  map will show all possible input combinations of the variables in the  form of a decimal number. For example this truth table shows the  possible combinations:

 This table shows the possible inputs for four different inputs, and  the last column, f, shows the value of the functions returned value for  each combination. In order to simplify this truth table, the map is  needed:

 It shows the same information but in a nicer, more concise format.  When an equation is applied, the sqaures will be filled with a output  value. 

Sheldon, Robert. “Karnaugh Map (K-map).” WhatIs.com, Aug. 2022, www.techtarget.com/whatis/definition/Karnaugh-map-K-map.

Storr, Wayne. “Laws of Boolean Algebra.” Basic Electronics Tutorials, Aug. 2022, www.electronics-tutorials.ws/boolean/bool_6.html 

 Links to an external site..

“De Morgan’s Law – Theorem, Statement, Proof, Examples.” Cuemath, www.cuemath.com/data/de-morgans-law.