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