Boolean Origination - History and Origin of Boolean Logic

Boolean logic or boolean algebra is a logical calculus which has only two values, 'true' or 'false'. It was conceived by mathematician, George Boole. During the time when Boole was defining his boolean logic, Charles Babbage was developing his 'analytical engine' - today's computer. So, the boolean logic has been in use with the ancestor of the digital computer. This logic is still an important tool used in search engines and databases for modern day operation. Here is a quick recap about the boolean origination, which describes the history and origin of boolean logic.

Flash Back!
In the year 1847, English mathematician George Boole (1815-1864) published, 'The Mathematical Analysis of Logic'. This book of his showed how using a specific set of logic can help one to wade through piles of data to find the required information. The importance of Boole's work was his way of approach towards logic. By incorporating logic into mathematics, Boole was able to determine what formed the base of boolean logic or algebra. It was the analogy which algebraic symbols had with those that represented logical forms. This basic analogy gave birth to what is known as the Boolean Logic or the Boolean Algebra. As we know that working of computers are based on the binary number system (1 or 0), where 1 means 'ON' and 0 signifies 'OFF'. These two states are represented by a difference in voltage. Now, the application of the boolean logic to the computer's binary number system was incorporated by an MIT grade student Claud Shannon. This was how the Boolean search came into place.

Boolean Logic Symbols
Precisely, Boolean Logic is defined as a logical system of operators - 'AND', 'OR' and 'NOT', and is a way of comparing individual bits. These connectors or operators are now used in computer construction, switching circuits, etc.

The AND, OR and NOT operators are also known as logic gates, and are used in logical operation. Their schematic diagram can be viewed from any book based on Boolean Algebra. The following describes the symbols and the operation of the logic gates.

The AND Gate
The AND gate is denoted by a dot (.). In an AND gate, there will be more than one input and only one output. Here, if all inputs are ON, the output will also be ON. And, if either of the inputs is OFF, then the output will also be OFF. The AND gate's symbol is '&'. Lets see the working in an example.

A . B = C (Here, A and B are the inputs, and C is the output)
As we know that in the binary number system, 1 means ON and 0 means OFF. So, if we take the inputs to be 1, the output will also give us 1.

A . B = C
1 . 1 = 1 (A = 1, B = 1).

If any of the input is taken as 0, then output will also be 0
A . B = C
1 . 0 = 0 (A = 1, B = 0)

The OR Gate
The OR gate is denoted by plus (+). Here, there will be more than one input and just one output. If we take both the inputs as 1, the output will also be 1. However, unlike the AND gate, if either of the inputs is 0, the output will still be one. Its symbol is '/'. Example;

A + B = C (Here, A and B are the inputs, and C is the output)

For A = 1, B = 1
A + B = C
1 +1 = 1

For A = 1, B = 0
A + B = C
1 + 0 = 1

For A = 0, B = 0
A + B = C
0 + 0 = 0

The NOT Gate
The NOT Gate is also known as the inverter gate. As the name suggests, here the output will be opposite to the input. There will be one input and one output. That is, if the input is 1 (ON), then the output will be 0 (OFF). The NOT gate is symbolized by a line over top of the input (Ā). The sign is also known as a 'complement'. For example,

For example, if A is the input, the the output will be Ā
That is,
For A = 1, output is 0
And for A = 0, ouput is 1

The NAND and the NOR gates are known to be the universal gates in boolean logic. Their combinations may be used to form any kind of logic gates. A NAND gate is formed by combining a NOT and AND gate. A NOR gate is a combination of a NOT and OR gate. The other logic gates are XOR (exclusive OR) and XNOR gates.

Simplifying Boolean Expressions
For simplifying boolean expressions, there are certain laws which need to be followed. They are the Boolean Rules for Simplification.

The Idempotent Laws
A . A = A
A + A = A

The Associative Laws
(A . B) C = A (B . C)
A + B) + C = A + (B + C)

The Commutative Laws
A . B = B . A
A + B = B + A

The Distributive Laws
A (B + C) = AB + AC
A + BC = (A + B) (A + C)

The Complement Laws
A . Ā = 0
A + Ā = 1

The Involution Law
A(double complement) = A

The Law of Union
A + 1 = 1
A + 0 = A

The Law of Intersection
A . 1 = A
A . 0 = 0

The Law of Absorption
A (A + B) = A
A + (A . B) = A

The Law of Common Identities
A (Ā + B) = A . B
A + (Ā . B) = A + B

DeMorgan's Law
(A . B)(complement) = A(complement) + B(complement)
(A + B)(complement) = A(complement) . B(complement)

Lets take a simple example of a simplified Boolean Expression. Suppose a logical circuit gives an expression A + ĀB = A + B
Now, the simplification goes like this,

A + ĀB
= (A + Ā) (A+B)
= A + B

With the help of truth table for each logic gates, and the logical circuit, it becomes easy to simplify any boolean expression.

The history of Boolean origination, has far more depths than what has been described. The Boolean logic has given way to several innovations and applications in the filed of computer science. No doubt, the Boolean logic or algebra is an admirable finding by one of the greatest mathematicians George Boole.