# The Brilliantly Interesting History And Origin of Boolean Logic

Boolean logic, also known as Boolean algebra, is defined to be a complete system for logical operations. This article gives you all the basic information regarding the history and origin of Boolean logic.

Rajib Singha

Last Updated: Feb 26, 2018

History

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 it into mathematics, Boole was able to determine what formed the base of Boolean algebra. It was the analogy which algebraic symbols had with those that represented logical forms.

The Symbols

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 '&'. Let's 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)

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

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 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. 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 gates are XOR (exclusive OR) and XNOR gates.

For example, if A is the input, 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. 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 gates are XOR (exclusive OR) and XNOR gates.

Simplifying Boolean Expressions

For simplifying Boolean expressions, there are certain laws which need to be followed.

The Idempotent Laws

A . A = A

A + A = A

A + A = A

The Associative Laws

(A . B) C = A (B . C)

A + B) + C = A + (B + C)

A + B) + C = A + (B + C)

The Commutative Laws

A . B = B . A

A + B = B + A

A + B = B + A

The Distributive Laws

A (B + C) = AB + AC

A + BC = (A + B) (A + C)

A + BC = (A + B) (A + C)

The Complement Laws

A . Ā = 0

A + Ā = 1

A + Ā = 1

The Involution Law

A(double complement) = A

The Law of Union

A + 1 = 1

A + 0 = A

A + 0 = A

The Law of Intersection

A . 1 = A

A . 0 = 0

A . 0 = 0

The Law of Absorption

A (A + B) = A

A + (A . B) = A

A + (A . B) = A

The Law of Common Identities

A (Ā + B) = A . B

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)

Let's 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.

(A + B)(complement) = A(complement) . B(complement)

Let's 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.