109 LearnersLast updated on June 12th, 2025
De Morgan's Law
De Morgan’s Law is a pair of rules in Boolean algebra, logic, and set theory. It relates the intersection and union of sets through complements. De Morgan’s Law is applied in the fields of mathematics, computer science, and design. In this article, we will learn more about De Morgan’s Law.
What is De Morgan’s Law
De Morgan’s Law explains how union, intersection, and complements are connected in set theory. In Boolean algebra, it shows how AND, OR, and NOT interact with each other. In logic, it shows the relationship between AND, OR, and NEGATION of the statement. De Morgan’s Law helps in optimizing various Boolean expressions and circuit designs.
De Morgan’s Laws:
In Logic (Boolean Algebra & Logic):
The negation of a disjunction (OR) is the conjunction (AND) of the negations:
¬(A ∨ B) = ¬A ∧ ¬B
¬(A ∨ B) = ¬A ∧ ¬B
Negating an AND statement turns it into an OR statement with each term negated:
¬(A ∧ B) = ¬A ∨ ¬B
¬(A ∧ B) = ¬A ∨ ¬B
In Set Theory:
The complement of a union is the intersection of the individual complements:
(A ∪ B)’ = A’ ∩ B’
(A ∪ B)’ = A’ ∩ B’
The complement of an intersection is the union of the individual complements:
(A ∩ B)’ = A’ ∪ B’
(A ∩ B)’ = A’ ∪ B’
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What is De Morgan’s Law Statement?
De Morgan’s Law says that negating a combined expression distributes the negation to each individual part while switching the AND/OR operation.
De Morgan’s Law in Set Theory
De Morgan’s Law in set theory describes how complements interact with union (OR) and intersection (AND). These laws help simplify expressions in probability and logic.
There are two laws in Set theory:
First law: Complement of Union
(A ∪ B)’ = A’ ∩ B’
The complement of the union is the intersection of their individual complements.
Second law: Complement of Intersection
(A ∩ B)’ = A’ ∪ B’
The complement of the intersection of two sets is the union of their individual complements.
De Morgan’s Law in Boolean Algebra
In Boolean algebra, De Morgan’s Law explains how the NOT operation interacts with AND and OR. These laws are crucial in designing logical circuits as they help simplify Boolean expressions.
The two laws in De Morgan’s Law state that:
The first law: Negation of OR
(A + B)’ = A’ . B’
This means that the negation or complement of A OR B is equal to NOT A AND NOT B.
The second law: The Negation of AND
(AB)’ = A’ + B’
This means that the negation or complement of A AND B is equal to NOT A OR NOT B.
De Morgan’s Law is used in Boolean algebra because it simplifies Boolean expressions and makes conversions between NAND and NOR gates easy.
What is De Morgan’s Law Formula
Let us learn more about De Morgan’s formulas in set theory, Boolean algebra, and logic.
In set theory:
- (A ∪ B)’ = A’ ∩ B’ ( The complement of the union of two sets is the intersection of their complements.)
- (A ∩ B)’ = A’ ∪ B’ (The complement of the intersection of two sets is the union of their complements.)
Where:
- ∪ = Union (OR in sets)
- ∩ = Intersection (AND in sets)
- ’ = Complement (NOT in sets)
In Logic:
- ~ (a ∧ b) ≡ ~ a ∨ ~ b (The negation of “A AND B” is equivalent to “NOT A OR NOT B”)
- ~ (a ∨ b) ≡ ~ a ∧ ~ b (The negation of “A OR B” is equivalent to “NOT A AND NOT B.”)
Where:
- ∼ → NOT (negation)
- ∧ = AND
- ∨ = OR
Common Mistakes and How to Avoid Them in De Morgan’s Law
De Morgan’s Law can be quite overwhelming as there are many rules to remember. This may lead to students making mistakes. Here are a few common mistakes students make and ways to avoid them:
Mistake 1
Incorrectly swapping AND (∧) and OR (∨).
When writing De Morgan’s Law, students must remember to write the correct logical expression (AND, OR) and apply the negation properly. Writing incorrectly would completely change the expression and violate the law.
Mistake 2
Not distributing the negation
Students must correctly apply negation to each term in the expression. Failing to do so would result in errors.
Mistake 3
Getting confused with the set theory
Students must remember to swap the union and intersections correctly when solving for questions in set theory. Students must also remember to complement each term individually.
Mistake 4
Errors when solving Boolean Algebra and Digital Logic
When simplifying logic circuits, students might incorrectly replace a NAND gate with an AND gate instead of a NOR gate.
Mistake 5
Applying De Morgan’s Law when not needed
Students may sometimes apply De Morgan’s law when there is no negation present.
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Real-life applications
De Morgan’s law is widely used in several fields mainly computer science. Here are some real-world uses of De Morgan’s Law:
- Circuit design: Engineers use De Morgan’s Law to simplify AND-OR logic and convert it into equivalent NAND or NOR circuits.
- Computer programming: Logical conditions used in programming can be rewritten to make the code more efficient.
- Search engines: De Morgan’s Law helps simplify complex search conditions for better accuracy.
Solved Examples on De Morgan’s Law
Problem 1
Simplify the logical expression ¬ ( A ∧ B ).
¬ A ∨ ¬ B
Explanation
De Morgan’s Law states that the negation of an AND is equivalent to the OR of the negations: ¬ (A ∧ B) = ¬ A ∨ ¬ B.
Problem 2
Express the complement of the intersection of sets A and B in terms of Ac and Bc
(A ∩ B )c = Ac ∪ Bc.
Explanation
De Morgan’s Law states that the complement of an intersection is the union of the complements.
Problem 3
Express the complement of the union of sets A and B in terms of A^c and B^c
(A∪B)c = Ac ∩ Bc.
Explanation
In sets, the complement of a union is given by the intersection of the complements.
Problem 4
Express the complement of the union of sets A, B, and C in terms of their complements.
(A ∪ B ∪ C)c = Ac ∩ Bc ∩ Cc.
Explanation
De Morgan’s law generalizes to multiple sets: the complement of a union equals the intersection of the complements.
Problem 5
Simplify the Boolean expression ¬((X ∧ Y) ∨ Z).
( ¬ X ∨ ¬ Y) ∧ ¬ Z.
Explanation
First, apply De Morgan’s Law to the outer OR:
¬((X ∧ Y) ∨ Z) = ¬(X ∧ Y) ∧ ¬Z ¬((X ∧ Y) ∨ Z) = ¬(X ∧ Y) ∧ ¬Z.
Then, simplify ¬(X ∧ Y) = ¬X ∨ ¬Y ¬(X ∧ Y) = ¬X ∨ ¬Y.
Thus the final result is: ( ¬ X ∨ ¬ Y) ∧ ¬ Z.
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FAQs On De Morgan’s Law
1.How do we use De Morgan’s Law in logic?
2.How is De Morgan’s Law useful in set theory?
3.Why is De Morgan’s Law important in programming?
4.Who was the founder of De Morgan’s Law?
5. Does De Morgan’s Law have any exceptions?
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Jaipreet Kour Wazir
About the Author
Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref
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: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!
