Introduction:

Boolean Algebra is a mathematical language of digital systems.

To design digital logic systems, the study of Boolean Algebra is imperative. It is one of the primitive tools for a logic designer, to design, analyze and simplify the logic. 

Outline:

  • To learn some basic rules, laws and theorems of Boolean Algebra
  • How to find the complement of a function
  • Examples and explanation
  • Applications of Boolean Algebra in digital electronics Or why do we need to learn Boolean Algebra

Three Basic Gates | Three Basic Operations In Boolean Algebra:

Boolean Inversion | The NOT Operator: 

It performs an inversion operation. The output is the inverse of the input. Get to know about NOT operators, please check out my tutorial here.

Boolean Multiplication | The AND Operator: 

It performs a Boolean multiplication operation. The laws and rules for multiplication are similar to elementary algebra. Get to know about NOT operators, please check out my tutorial here.

Boolean Addition | The OR Operator: 

Boolean addition is equivalent to the OR operation 

Laws Of Boolean Algebra:

The fundamental laws of arithmetic are also valid here. These laws are here 

  • Associative law
  • Commutative law
  • Distributive law

Commutative Law:

Commutative law for addition:

Commutative law for two variables is given below:

A + B = B + A

Commutative law for multiplication:

Commutative law for two variables is given below:

A . B = B . A

Associate Law:

Associative law for addition:

The associative law for three variables is given below:

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

Associative law for multiplication:

The associative law for three variables is given below:

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

Distributive Law:

The distributive law for three variables is given below:

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

Rules Of Boolean Algebra:

For practical digital systems, many complex logic blocks appear during the analysis. There are some rules of solving and simplifying logic expressions, which are almost the same as that of arithmetic rules. The topic is rather easy as the rules and laws are almost the same. Let’s have a look at them.

A + 0 = A

This is the additive identity. 

For a two-input OR gate, one input is zero and the variable A has only two values (either 1 or 0). The output is equal to A. 

To prove this rule, let’s have a look at the truth table.

A2nd InputOutput
000
101

2. A + 1 = 1:

= (A + 1) . 1    Rule 4

= (A + 1) . (A + AC). Rule 6

= (A + 1 . AC) Rule 12

= (A + AC) Rule 4

= 1 Rule 6

A . 0 = 0

This is a duality property.

A . 1 = A

This is multiplicative identity. Have a look at Rule 1, we will prove this rule in the same manner. 

A + A = A

Proof:

A + A = (A + A) . 1   Rule 4

 = (A + A) . (A + AC ) Rule 6

 = A + A . ARule 12

 = A + 0 Rule 8

 = A    Rule 4

A + AC = 1

This rule is verified with the help of a truth table. The right-hand side is equal to the left-hand side for both values of A.

AACOutput = 1
011
101

A . A = A

Proof:

A . A = A . A + 0   Rule 1

= A . A + A . AC Rule 8

= A . (A + AC) Distributive Law

= A . 1 Rule 6

= A 

A . AC = 0

This rule is verified with the help of a truth table. 

AACOutput = 0
010
100

(AC)C = A

Double negation or double inversion. It is easily seen in schematic diagram.

  1. A + A . B = A

= A . 1 + A . B Rule 4

= A . (1 + B) Distributive Law

= A . 1 Rule 2

= A Rule 4

A + ACB = A + B

= (A + A . B) + AC.BRule 10

= A + B . (A + AC)  Distributive Law

= A + B . (1) Rule 6

= A + B Rule 4

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

= A . A + A . C + A . B + B .C  Distributive law

= A + A . C + A . B + B .C  Rule 7

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

Distributive law

= A . 1 + A . B + B .C Rule 2

= A . (1 + B) + B . C Distributive law

= A .1 + B . C Rule 2 

= A + B . C Rule 4

Here some of more Boolean Expression exercises.

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