George Boole (1815-64) was a British mathematician who developed the laws of Boolean algebra, which are fundamental to the workings of modern computers. Read the Wikipedia article here.

Boolean algebra is a method of performing logic calculations using variables that contain the boolean values true and false. By assigning the value 1 to true and 0 to false boolean algebra can be used to perform boolean logic calculations and to display logic circuits in a simple mathematical form. There are several basic laws that form the axioms (basic absolute rules/principles) of boolean algebra. From these axioms even more complex rules can be formed.

But before you learn the axioms of boolean algebra you need to learn the notation:

0 - Binary bit equal to false in terms of logic

1 - Binary bit equal to true in terms of logic

A - Pronumerals are used to represent variable bits such as inputs or the output of gates

Ā - A line on top of a variable represents that the variable has been through a NOT gate. multiple lines mean its been through multiple NOT gates. This line stretches across the entire statement entering the gate

+ - Plus signs represent an OR gate. In boolean algebra the line 'A+1' is read A OR 1

• - Dot multiplication symbols are used to represent AND gates. The line A•B is read A AND B. Its not necessary to use this symbol and A•B may be represented as AB instead

ꚛ - This symbol is used to represent a XOR gate.

ꙩ - While this symbol will not be used in this course it represents an XNOR gate. Instead a XOR gate expression with a line above it will be used

() - Parentheses are used to represent grouping in the same way they do in ordinary mathematics

Now on to the various axioms of boolean algebra shown here:

One thing you may have noticed is that the rules on one side of the label is the same as the rule on the other side but with their ORs and ANDs switched. This is referred to as the duality of boolean algebra and each formula happens to be the conjugate dual of the formula on the other side of the label.

The rules for duality are:

• Swap all AND and OR gates

• Invert all 1s and 0s

• Maintain all parentheses and variables names/states

Using the axioms it is possible it is possible to construct more complicated rules. Then by applying the duality rules you can find the conjugate dual rule for it. From here I will list more complicated rules and their proofs.

Simplifying Logic Circuits using Boolean Algebra

The laws of boolean algebra can be applied to logic circuits to simplify them. Their are two ways to go about this one involves applying these laws to