De Morgan's Laws

In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference.

Version

v1 · 2026-09-28 · History

Domain-specific #

8872

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Mathematical Logic, Boolean Algebra → Mathematics

Core Idea

De Morgan's Laws is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19^th-century British mathematician.

How would you explain it like I'm…

Flip-the-Not Rules

If it's NOT true that 'you have a cat AND a dog,' then you're missing at least one: no cat, or no dog. And if it's NOT true that 'you have a cat OR a dog,' then you have no cat AND no dog. De Morgan's laws are these two rules for flipping 'not' through 'and' and 'or.'

Flipping And and Or With Not

De Morgan's Laws are two rules in logic about the words 'and', 'or', and 'not'. The first says: 'not (A and B)' means the same as 'not A, or not B.' The second says: 'not (A or B)' means the same as 'not A, and not B.' So when you put 'not' in front of a grouped statement, the 'and' flips to 'or', or the 'or' flips to 'and', and each part gets its own 'not.' They're named after Augustus De Morgan, a British mathematician from the 1800s.

Negation Duality Laws

De Morgan's laws are a pair of transformation rules in propositional logic and Boolean algebra, and both are valid rules of inference. They state that the negation of a conjunction equals the disjunction of the negations, ¬(A ∧ B) ≡ ¬A ∨ ¬B, and the negation of a disjunction equals the conjunction of the negations, ¬(A ∨ B) ≡ ¬A ∧ ¬B. In words: "not (A and B)" is "not A or not B," and "not (A or B)" is "not A and not B." This means "and" and "or" can each be expressed in terms of the other using negation. A common mistake they help avoid is thinking "not (A and B)" means "not A and not B." They are named after the 19th-century British mathematician Augustus De Morgan.

 

De Morgan's laws, also called De Morgan's theorem, are a pair of transformation rules in propositional logic and Boolean algebra, each of which is a valid rule of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The first law states that the negation of a conjunction is equivalent to the disjunction of the negations: ¬(A ∧ B) ≡ ¬A ∨ ¬B. The second states that the negation of a disjunction is equivalent to the conjunction of the negations: ¬(A ∨ B) ≡ ¬A ∧ ¬B. Together they show that conjunction and disjunction can each be expressed purely in terms of the other plus negation, a duality that is used when simplifying Boolean expressions or logic circuits. Because they are valid rules of inference, either side may be replaced by the other in any derivation. The identity of the concept is this specific pair of equivalences, not negation or Boolean algebra in general.

Scope of Application

• In computer engineering. De Morgan's laws are widely used in computer engineering and digital logic for the purpose of simplifying circuit designs.

• Generalising De Morgan duality. The existence of negation normal forms drives many applications, for example in digital circuit design, where it is used to manipulate the types of logic gates, and in formal logic, where.

• Worked Example. Here is a more concrete example to illustrate how De Morgan's laws operate in practice.

• Negation of a disjunction. In the case of its application to a disjunction, consider the following claim: "it is false that either of A or B is true", which is written as.

• Negation of a conjunction. The application of De Morgan's theorem to conjunction is very similar to its application to a disjunction both in form and rationale.

Clarity

A clear use of De Morgan's Laws names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference.

Manages Complexity

De Morgan's Laws compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—this transformation demonstrates how negation is distributed across a conjunction by negating each component and switching the logical operator.—and the practical consequence—the proof that \overline{A\cap B} = \overline{A} \cup \overline{B} is completed in 2 steps by proving both \overline{A\cap B} \subseteq.

Abstract Reasoning

1. Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
2. State the relation. Use the source-grounded identity: In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference.
3. Check operation and conditions. De Morgan's formulation was influenced by the algebraization of logic undertaken by George Boole, which later cemented De Morgan's claim to the find.
4. Demand recognition evidence.

Knowledge Transfer

Within the home domain. Knowledge about De Morgan's Laws transfers literally when a new case preserves the same carrier type, relation, and recognition test. De Morgan's laws are widely used in computer engineering and digital logic for the purpose of simplifying circuit designs. The existence of negation normal forms drives many applications, for example in digital circuit design, where it is used to manipulate the types of logic gates, and in formal logic, where it is needed to find the conjunctive normal form and disjunctive normal form of a formula. Beyond the home domain.

Neighborhood in Abstraction Space

De Morgan's Laws sits in a moderately populated region (44^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Formal Logic & Semantic Systems (18 abstractions)

Nearest neighbors

• Square of opposition — 0.88
• Computability logic — 0.87
• False position method — 0.87
• Categorial Grammar — 0.87
• Typographical Number Theory — 0.86

Computed from structural-signature embeddings · 2026-10-08