cancellation law: meaning, definition, pronunciation and examples
C2Technical / Academic
Quick answer
What does “cancellation law” mean?
A mathematical principle stating that if an operation performed on two equal quantities yields the same result, then the original quantities were equal. Formally, if a · c = b · c, then a = b (for multiplication), or if a + c = b + c, then a = b (for addition), provided the operation is defined and 'c' is not a special element (like zero for multiplication in some contexts).
Audio
Pronunciation
Definition
Meaning and Definition
A mathematical principle stating that if an operation performed on two equal quantities yields the same result, then the original quantities were equal. Formally, if a · c = b · c, then a = b (for multiplication), or if a + c = b + c, then a = b (for addition), provided the operation is defined and 'c' is not a special element (like zero for multiplication in some contexts).
A principle in logic, algebra, and set theory allowing a common term on both sides of an equation or relation to be removed, simplifying the statement. The concept extends to abstract algebraic structures like groups and monoids, where it defines properties of the operation. In everyday language, it can metaphorically refer to a rule or situation where two things are deemed equivalent because they lead to the same outcome.
Dialectal Variation
British vs American Usage
Differences
No significant difference in meaning or usage. Spelling of 'cancellation' follows regional norms: British English often uses 'cancellation' (double 'l'), while American English accepts both 'cancellation' and 'cancelation', though 'cancellation' remains standard in formal and mathematical texts.
Connotations
None beyond the technical meaning.
Frequency
Equally rare and specialised in both varieties, confined to mathematics, computer science, and formal logic contexts.
Grammar
How to Use “cancellation law” in a Sentence
The [operation] satisfies the cancellation law.The cancellation law holds for [set/structure] under [operation].By the cancellation law, we deduce that [result].Vocabulary
Collocations
Examples
Examples of “cancellation law” in a Sentence
verb
British English
- You can cancel the common factor here because the cancellation law holds.
- We cancelled the term using the left cancellation law.
American English
- You can cancel the common factor here because the cancellation law holds.
- We canceled the term using the left cancellation law.
adverb
British English
- The elements cancellationally simplify the equation.
American English
- The elements cancellationally simplify the equation.
adjective
British English
- A cancellation-law property is fundamental.
- The monoid is cancellation-law-abiding.
American English
- A cancellation-law property is fundamental.
- The monoid is cancellation-law-abiding.
Usage
Meaning in Context
Business
Not used.
Academic
Core term in undergraduate abstract algebra, number theory, and formal logic courses. Used in proofs and definitions of algebraic structures like integral domains.
Everyday
Virtually never used. Potential metaphorical use: 'In their relationship, the cancellation law applied—if an argument started with both being tired, it didn't matter who began it.'
Technical
Precise term in mathematics, computer science (especially in theory of semigroups or monoids), and mathematical logic papers.
Vocabulary
Synonyms of “cancellation law”
Neutral
Weak
Vocabulary
Antonyms of “cancellation law”
Watch out
Common Mistakes When Using “cancellation law”
- Using it to refer to a policy about cancelling bookings or events.
- Applying it incorrectly in algebraic structures where it does not hold (e.g., matrix multiplication without invertibility conditions).
- Confusing left and right cancellation laws in non-commutative contexts.
FAQ
Frequently Asked Questions
Yes, the informal process of 'cancelling out' a common term from both sides of an equation is an application of the formal cancellation law, provided the operation and the term being cancelled satisfy the necessary conditions (e.g., the term is not zero in multiplication).
Left cancellation law: if c · a = c · b, then a = b. Right cancellation law: if a · c = b · c, then a = b. In commutative operations (like standard addition/multiplication), they are equivalent. In non-commutative operations (like matrix multiplication), they are distinct properties.
Not directly as a 'law' in the same sense. Subtraction is the inverse of addition, and 'cancelling' in subtraction is actually adding the same value to both sides, which is justified by the additive cancellation law. Division is similar for multiplication. The core laws are defined for the primary operations (addition, multiplication).
It is a key property used to define and classify algebraic structures. Structures that satisfy cancellation laws (cancellative semigroups or monoids) have useful properties. The failure of the cancellation law in a ring indicates the presence of zero divisors, which is a crucial distinction (e.g., between integral domains and other rings).
A mathematical principle stating that if an operation performed on two equal quantities yields the same result, then the original quantities were equal. Formally, if a · c = b · c, then a = b (for multiplication), or if a + c = b + c, then a = b (for addition), provided the operation is defined and 'c' is not a special element (like zero for multiplication in some contexts).
Cancellation law is usually technical / academic in register.
Cancellation law: in British English it is pronounced /ˌkænsəˈleɪʃən ˌlɔː/, and in American English it is pronounced /ˌkænsəˈleɪʃən ˌlɔ/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think of cancelling a common factor on both sides of an equation, like crossing out the same item on two identical shopping lists. If the lists are equal after removing that item, the remaining items must have been equal too.
Conceptual Metaphor
MATHEMATICAL PRINCIPLES ARE LAWS; SIMPLIFICATION IS CANCELLATION.
Practice
Quiz
In which of these algebraic structures does the cancellation law for multiplication NOT necessarily hold?