Boolean Algebra in Digital Electronics

Boolean algebra is a set of elements, operators, and some number of unproven axioms or postulates.

Basically, it is used to analyze and simplify the digital circuit. 0 and 1 are the only two numbers used here, and these two numbers are also called binary numbers.

Boolean algebra was invented in 1954 by George Boale.

Symbols used in boolean algebra such as x, y, a, b, etc. do not represent any numerical values. Only two possible values are allowed here, which are 0 and 1.

Addition Rule in Boolean Algebra

The table given below shows the addition rule used in boolean algebra:

Input 1 Operator Input 2 Operator Result/Output
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 1

Or you can also write the addition rule without a table, as given below:

0+0=0
0+1=1
1+0=1
1+1=1

It functions similarly to an OR gate. That is, if any one of the inputs is 1, then the output will be 1, otherwise the output will be 0.

Multiplication Rule in Boolean Algebra

The table given below shows the multiplication rule used in boolean algebra:

Input 1 Operator Input 2 Operator Result/Output
0 . 0 = 0
0 . 1 = 0
1 . 0 = 0
1 . 1 = 1

Or you can also write the multiplication rule without using a table, just like this, as given below:

0.0=0
0.1=0
1.0=0
1.1=1

The preceding rule is based on the AND Gate's rule.That is, the output will be 1 if and only if all the input is 1, otherwise the output or result will be 0.

Basic Laws of Boolean Algebra

There are three basic laws of Boolean algebra, as given below:

  1. Commutative Law
  2. Associative Law
  3. Distributive Law

Now let's briefly describe all three of the above basic laws of Boolean algebra one by one.

Commutative Law

There are two laws available in this category, as given below:

  1. Commutative law of addition
  2. Commutative law of multiplication

We're talking about laws with two variables here.

The commutative law of addition for two variables is algebraically expressed as:

A+B = B+A

And the commutative law of multiplication for two variables is algebraically expressed as:

A.B = B.A

After determining the above two types of commutative laws, we have concluded that, on changing the sequence of variables or inputs, the output does not get affected by any logic circuit in cases where these two laws are applied.

Associative Law

There are also two types of laws available in this category, namely:

  1. Associative law of addition
  2. Associative law of multiplication

Now, here we are talking about laws over three variables, say, A, B, and C.

The associative law of addition of three variables is algebraically expressed as:

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

And the associative law of multiplication of three variables is algebraically expressed as:

A.(BC) = (AB).C

Distributive Law

In boolean algebra, the distributive law of three variables is algebraically expressed as:

A.(B+C) = AB+AC

Theorems and Postulates of Boolean Algebra

As we all know, theorems are true statements, and all the statements can be proven, but postulates are all those statements that are assumed to be true without proof. And theorems can be proven with the help of available postulates.

Therefore, we have divided a total of 17 algebraic equations into two categories, that is, theorems and postulates. In theorems, we have a total of nine algebraic equations, and in postulates, we have a total of eight algebraic equations.

But here we will prove all 17 algebraic equations with the help of the truth table.

Theorems of Boolean Algebra

Here we will talk about a total of 5 theorems (a total of 9 algebraic equations) available in boolean algebra, as given in the following table:

Theorem Equation
Theorem 1 over addition X+X = X
Theorem 1 over multiplication X.X = X
Theorem 2 over addition X+1 = 1
Theorem 2 over multiplication X.0 = 0
Involution Theorem (X')' = X
Associativity theorem 1 X+(Y+Z) = (X+Y)+Z
Associativity theorem 2 (X.Y)' = X'+Y'
Absorption theorem 1 X+XY = X
Absorption theorem 2 X(X+Y) = X

Postulates of Boolean Algebra

There are a total of four postulates available in boolean algebra that we will discuss. In each postulate, there are two algebraic equations, giving a total of eight algebraic equations.

All four postulates (eight algebraic equations) are listed in the table below:

Postulate Equation
Postulate 1 over addition X+0 = X
Postulate 1 over multiplication X.1 = X
Postulate 2 over addition X+X' = 1
Postulate 2 over multiplication X.X' = 0
Commutativity postulate 1 X+Y = Y+X
Commutativity postulate 2 X.Y = Y.X
Distributivity postulate 1 X.(Y+Z) = XY+XZ
Distributivity postulate 2 X+YZ = (X+Y)(X+Z)

The proofs of all the above theorems and postulates are given in the next tutorial or article.

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