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de morgan's laws

C1/C2
UK/də ˈmɔːɡənz lɔːz/US/də ˈmɔːrɡənz lɑːz/

Formal, Academic, Technical

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Definition

Meaning

A pair of logical equivalences from propositional logic and set theory that describe how negation interacts with conjunction (AND) and disjunction (OR).

In set theory: the complement of a union is the intersection of the complements, and the complement of an intersection is the union of the complements. In logic: not (P and Q) is equivalent to (not P) or (not Q), and not (P or Q) is equivalent to (not P) and (not Q).

Linguistics

Semantic Notes

Always plural ('laws'), not 'law'. Refers to a specific, fixed theorem. Named after the mathematician Augustus De Morgan (19th century).

Dialectal Variation

British vs American Usage

Differences

No significant lexical differences. Possessive apostrophe usage is consistent ('De Morgan's'). The formal register eliminates regional variation.

Connotations

Strictly mathematical/logical connotations. No cultural or regional variation.

Frequency

Used exclusively in technical contexts (mathematics, computer science, logic). No frequency difference between regions.

Vocabulary

Collocations

strong
apply De Morgan's lawsusing De Morgan's lawsstate De Morgan's laws
medium
simplify using De Morgan's lawsa consequence of De Morgan's laws
weak
importantusefullogical

Grammar

Valency Patterns

[Subject] + applies + De Morgan's laws + to + [logical expression/set expression]

Vocabulary

Synonyms

Strong

the laws of complementation in Boolean algebra

Neutral

De Morgan's theorems

Weak

negation ruleslogical equivalences

Usage

Context Usage

Business

Rarely used outside data analytics or formal logic in systems design.

Academic

Core concept in discrete mathematics, logic, computer science, and set theory courses.

Everyday

Virtually never used in everyday conversation.

Technical

Fundamental tool for simplifying Boolean expressions in digital circuit design, programming, and formal proofs.

Examples

By Part of Speech

adjective

British English

  • De Morgan's principles are essential.

American English

  • The De Morgan transformation is useful.

Examples

By CEFR Level

B2
  • You can use De Morgan's laws to rewrite this logical condition in a simpler way.
  • The proof relied on an application of De Morgan's laws.
C1
  • By applying De Morgan's laws to the complement of the union, we obtained an intersection of complements, which was far easier to compute.
  • The engineer simplified the complex chip logic gate diagram by iteratively invoking De Morgan's laws.

Learning

Memory Aids

Mnemonic

Think 'Break the line, change the sign': When you distribute a negation over AND or OR, you break the line (of the operation) and change the sign (AND becomes OR, OR becomes AND).

Conceptual Metaphor

DISTRIBUTION OF NEGATION AS A DISTRIBUTIVE LAW.

Watch out

Common Pitfalls

Translation Traps (for Russian speakers)

  • Avoid translating as 'De Morgan's rights' (права). Use 'законы де Моргана'.
  • Be careful with the apostrophe and possessive 's' in transliteration.

Common Mistakes

  • Calling it 'De Morgan law' (singular).
  • Misapplying the law by forgetting to negate the individual components after changing the operator.
  • Writing 'Demorgan' as one word.

Practice

Quiz

Fill in the gap
To find the complement of A ∩ B, one must apply to get A' ∪ B'.
Multiple Choice

What is the correct application of De Morgan's law to ¬(P ∨ Q)?

FAQ

Frequently Asked Questions

No, they have direct analogues in set theory, relating the complement of unions and intersections.

In Boolean algebra, they are often considered foundational laws or identities, though they are proven theorems in logic and set theory.

Yes, but remembering one helps derive the other by duality (swap AND/OR and negate).

Most commonly in digital electronics, computer programming (optimising conditionals), and simplifying logical proofs.

de morgan's laws - meaning, definition & pronunciation - English Dictionary | Lingvocore