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.
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.
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.
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.
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.
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.
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
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
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 Laws are commonly used to simplify Boolean expressions, complement expressions, and convert between AND and OR operations. Let’s explain some examples.
De Morgan’s Law Example 1: (A · B)’
The first De Morgan’s Law can be used to change the complemented AND operation into an OR operation.
- (A · B)’ = A’ + B’
De Morgan’s Law Example 2: (A + B)’
The second De Morgan’s Law can be used to change the complemented OR operation into an AND operation.
- (A + B)’ = A’ · B’
De Morgan’s Law Example 3: (AB)’
The complement of an AND expression can be changed into an OR expression.
- (AB)’ = A’ + B’
De Morgan’s Law Example 4: (A + B + C)’
The complement of an OR expression becomes an AND expression with all variables complemented.
- (A + B + C)’ = A’B’C’
De Morgan’s Law Example 5: (ABC)’
The complement of an AND expression becomes an OR expression with all variables complemented.
- (ABC)’ = A’ + B’ + C’
De Morgan’s Law Example 6: (A + BC)’
The complement applies to the complete expression. First, the OR operation changes to AND, and each term is complemented.
- (A + BC)’ = A'(BC)’
- (A + BC)’ = A'(B’ + C’)
De Morgan’s Law Example 7: (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 8: (ABCD)’
The complement of an AND expression becomes an OR expression with all variables complemented.
- (ABCD)’ = A’ + B’ + C’ + D’.