Introduction to DLD

De Morgan’s Law in Boolean Algebra

According to De Morgan’s Law in Boolean Algebra, taking the complement of an AND or OR expression changes the operation and complements each Boolean variable. De Morgan’s Laws are used to simplify Boolean expressions and convert between AND and OR operations.

De Morgan’s Laws apply to both AND and OR operations.

  • The complement of an AND operation becomes an OR operation with complemented variables.
  • The complement of an OR operation becomes an AND operation with complemented variables.

De Morgan’s Laws are fundamental laws used to simplify Boolean expressions and understand the behavior of digital logic circuits.

Types of De Morgan’s Laws

There are two types of De Morgan’s Laws in Boolean algebra:

1. De Morgan’s First Law

De Morgan’s First Law states that the complement of an AND operation is equal to the OR operation of the complemented variables. It can be expressed as: (A · B)’ = A’ + B’

The following diagram of De Morgan’s First Law explains it.

De Morgans Law in Boolean Algebra - First De Morgans Law

Therefore, (A · B)’ = A’ + B’ because both expressions produce the same output. It is called De Morgan’s First Law.

2. De Morgan’s Second Law

De Morgan’s Second Law states that the complement of an OR operation is equal to the AND operation of the complemented variables. It can be expressed as: (A + B)’ = A’ · B’

The following diagram of De Morgan’s Second Law explains it.

De Morgans Law in Boolean Algebra - Second De Morgans Law

Therefore, (A + B)’ = A’ · B’ because both expressions produce the same output. It is called De Morgan’s Second Law.

De Morgan’s Law Truth Table

Let’s explain the truth table of the first and second De Morgan’s Laws in Boolean algebra.

First De Morgan’s Law – Truth Table

In De Morgan’s First Law, the complement of an AND operation is equal to the OR operation of the complemented variables. The following diagram shows the truth table of De Morgan’s First Law.

First De Morgans Law in Boolean Algebra - Truth Table

Second De Morgan’s Law – Truth Table

In De Morgan’s Second Law, the complement of an OR operation is equal to the AND operation of the complemented variables. The following diagram shows the truth table of De Morgan’s Second Law.

Second De Morgans Law in Boolean Algebra - Truth Table

De Morgan’s Law Logic Gates

Let’s explain the logic gates of De Morgan’s Laws in Boolean algebra.

First De Morgan’s Law – Logic Gate

In De Morgan’s First Law, the complemented AND operation is equivalent to an OR operation with complemented inputs. The following diagram shows De Morgan’s First Law using logic gates.

First ( De Morgans Law ) Logic Gate

The first circuit represents (A · B)’, while the second circuit represents A’ + B’. Both circuits produce the same output.

Second De Morgan’s Law – Logic Gate

In De Morgan’s Second Law, the complemented OR operation is equivalent to an AND operation with complemented inputs. The following diagram shows De Morgan’s Second Law using logic gates.

Second ( De Morgans Law ) Logic Gate

The first circuit represents (A + B)’, while the second circuit represents A’ · B’. Both circuits produce the same output.

De Morgan’s Law – Circuit Switches

Let’s explain De Morgan’s Laws using circuit switches. De Morgan’s Laws can be represented using circuit switches to show how complementing an AND or OR operation changes the circuit arrangement.

First De Morgan’s Law – Circuit Switches

In a series circuit, the AND operation requires all switches to be closed for the output to be 1. When the entire output is complemented, the result can be represented by switches in a parallel arrangement with complemented inputs.

The following diagram explains the process

First De Morgans Law - Circuit Switches

In the first circuit, A and B are connected in series and the output is complemented. In the second circuit, the complemented switches A’ and B’ are connected in parallel. Both circuits produce the same output.

  • Case 01: When A = 0 and B = 0, the output is 1.
  • Case 02: When A = 0 and B = 1, the output is 1.
  • Case 03: When A = 1 and B = 0, the output is 1.
  • Case 04: When A = 1 and B = 1, the output is 0.

Therefore, the two circuit arrangements produce the same output.

Second De Morgan’s Law – Circuit Switches

In a parallel circuit, the OR operation produces an output of 1 when at least one switch is closed. When the entire output is complemented, the result can be represented by switches in a series arrangement with complemented inputs.

The following diagram explains the process

Second De Morgans Law - Circuit Switches

In the first circuit, A and B are connected in parallel and the output is complemented. In the second circuit, the complemented switches A’ and B’ are connected in series. Both circuits produce the same output.

  • Case 01: When A = 0 and B = 0, the output is 1.
  • Case 02: When A = 0 and B = 1, the output is 0.
  • Case 03: When A = 1 and B = 0, the output is 0.
  • Case 04: When A = 1 and B = 1, the output is 0.

Therefore, the two circuit arrangements produce the same output.

De Morgan’s Law in Boolean Simplification

De Morgan’s Law is one of the Boolean algebra laws used to simplify Boolean expressions. Like other Boolean laws, it can be applied along with other laws to make expressions simpler and reduce the number of logic gates required in a digital circuit.

Let’s explain some examples where De Morgan’s law is used to simplify Boolean expressions

De Morgan’s Law – Example 1

In the first example, the Boolean expression (A+(B’.C))’ is given. The following diagram simplifies the Boolean expression using De Morgan’s law

De Morgans Law in Boolean Algebra - Example 1

In the above diagram,

  • First, apply De Morgan’s Law to change the complemented AND operation into an OR operation and complement each variable.
  • Then, apply the Double Negation Law to simplify the expression to A’ · (B + C’).

De Morgan’s Law – Example 2

In the second example, the Boolean expression ((A+C).(A+B’).(A’+B’.C’))’ is given. The following diagram simplifies the Boolean expression using De Morgan’s law

De Morgans Law in Boolean Algebra - Example 2

In the above diagram,

  • First, apply De Morgan’s Law
  • Then, apply the Double Negation Law to simplify the expression

De Morgan’s Law – Example 3

In the third example, the Boolean expression A.[B+C’.(AB+AC’)’] is given. The following diagram simplifies the Boolean expression using De Morgan’s law

De Morgans Law in Boolean Algebra - Example 3

De Morgan’s Law – Example 4

In the 4th example, the Boolean expression (A+(BC)’+ (AC)’)’ is given. The following diagram simplifies the Boolean expression using De Morgan’s law

De Morgans Law in Boolean Algebra - Example 4

 

De Morgan’s Law – Example 5

In the 5th example, the Boolean expression (A + (B.C.D)’ + (A’.B)’)’  is given. The following diagram simplifies the Boolean expression using De Morgan’s law

De Morgans Law in Boolean Algebra - Example 5

De Morgan’s Law – Example 6

In the 6th example, the Boolean expression ((A ⊕ B) . C’)’  is given. The following diagram simplifies the Boolean expression using De Morgan’s law

De Morgans Law in Boolean Algebra - Example 6

De Morgan’s Law – Example 7

In the 7th example, the Boolean expression [[(A+ (B.C)]’ + (A.B’)’]’  is given. The following diagram simplifies the Boolean expression using De Morgan’s law

De Morgans Law in Boolean Algebra - Example 7

De Morgan’s Law – Example 8

In the 8th example, the Boolean expression [(A.B)’ . (C’.D’) . (E.F)’]’  is given. The following diagram simplifies the Boolean expression using De Morgan’s law

De Morgans Law in Boolean Algebra - Example 8

De Morgan’s Law Example 9: (A + B + C + D)’

The complement of an OR expression becomes an AND expression with all variables complemented.

  • (A + B + C + D)’ = A’B’C’D’

De Morgan’s Law Example 10: (ABCD)’

The complement of an AND expression becomes an OR expression with all variables complemented.

  • (ABCD)’ = A’ + B’ + C’ + D’.