Boolean algebra and Logic gates

Boolean Algebra Basics

The mathematical framework of Boolean algebra deals with binary variables and logical procedures. George Boole invented it in the mid-nineteenth century, and it is widely used in digital electronics and computer science. Variables in Boolean algebra can only have two values: true (expressed as 1) or false (represented as 0).

Boolean Operators:

AND (·): When both input variables are true, the AND operator returns true (1); otherwise, it gives false (0).

Truth Table:

A  |  B  | A AND B

------------------

0  |  0  |   0

0  |  1  |   0

1  |  0  |   0

1  |  1  |   1

OR (+): When at least one of the input variables is true, the OR operator gives true (1); otherwise, it returns false (0).

 

Truth Table:

A  |  B  | A OR B

-----------------

0  |  0  |   0

0  |  1  |   1

1  |  0  |   1

1  |  1  |   1

NOT (¬): The NOT operator (also known as the inversion or complement operator) negates the input variable; if it is true (1), it returns false (0), and vice versa.

 

Truth Table:

A  | NOT A

---------

0  |   1

1  |   0

 

Boolean Laws:

Commutative Law: The sequence of variables in an operation has no effect on the outcome.

A AND B = B AND A

A OR B = B OR A

Associative Law: It states that grouping variables in an operation has no effect on the outcome.

(A AND B) AND C = A AND (B AND C)

(A OR B) OR C = A OR (B OR C)

Distributive Law: Boolean multiplication distributes more than Boolean addition (and vice versa).

A AND (B OR C) = (A AND B) OR (A AND C)

A OR (B AND C) = (A OR B) AND (A OR C)

Identity Law: AND has an identity element of 1 (true), while OR has an identity element of 0 (false).

A AND 1 = A

A OR 0 = A

Annihilation Law: For AND, the annihilator element is 0 (false), while for OR, it is 1 (true).

A AND 0 = 0

A OR 1 = 1

 

Logic Gates and their Truth Tables

Physical objects or electrical circuits that implement Boolean operations are known as logic gates. Based on their truth tables, they take binary inputs and make binary outputs. Among the most frequent logic gates are:

AND Gate:

Truth Table:

A  |  B  | A AND B

------------------

0  |  0  |   0

0  |  1  |   0

1  |  0  |   0

1  |  1  |   1

 

OR Gate:

Truth Table:

A  |  B  | A OR B

-----------------

0  |  0  |   0

0  |  1  |   1

1  |  0  |   1

1  |  1  |   1

 

NOT Gate:

Truth Table:

A  | NOT A

---------

0  |   1

1  |   0

 

NAND Gate:

Truth Table:

A  |  B  | A NAND B

------------------

0  |  0  |    1

0  |  1  |    1

1  |  0  |    1

1  |  1  |    0

 

NOR Gate:

Truth Table:

A  |  B  | A NOR B

-----------------

0  |  0  |   1

0  |  1  |   0

1  |  0  |   0

1  |  1  |   0

 

Implementing Logic Functions using Gates

Combinations of simple logic gates can be used to build complex logic functions. The logic function F = AB + C'D, for example, can be implemented with AND, OR, NOT, and NAND gates. 

NAND/NOR logic and universal gates

NAND and NOR gates are regarded as "universal gates" in digital electronics because they may be used to build any other logic gate. That is, any logic function may be written using only NAND or NOR gates.

Implementing AND using NAND:

  A  |  B  | NAND  | AND

  ----------------------

  0  |  0  |   1   |  0

  0  |  1  |   1   |  0

  1  |  0  |   1   |  0

  1  |  1  |   0   |  1

 

Implementing OR using NOR:

  A  |  B  | NOR  | OR

  --------------------

  0  |  0  |  1   |  0

  0  |  1  |  0   |  1

  1  |  0  |  0   |  1

  1  |  1  |  0   |  1

 

Because of their simplicity and versatility, universal gates such as NAND and NOR are frequently employed in digital circuit design. They can reduce the amount of components needed in a design and simplify the overall complexity.