surjection: meaning, definition, pronunciation and examples
Very lowTechnical/Formal
Quick answer
What does “surjection” mean?
A mathematical function where every element in the codomain is mapped to by at least one element in the domain.
Audio
Pronunciation
Definition
Meaning and Definition
A mathematical function where every element in the codomain is mapped to by at least one element in the domain.
In mathematics, a surjective function (onto function) ensures complete coverage of the target set; in broader metaphorical use, it can describe any mapping or relationship that comprehensively covers all targets.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage. Spelling and pronunciation follow standard regional patterns.
Connotations
Purely technical term with no regional connotative differences.
Frequency
Equally rare in both varieties, confined to technical contexts.
Grammar
How to Use “surjection” in a Sentence
[be] a surjection[define/prove/show] surjectionsurjection from X to Ysurjection onto YVocabulary
Collocations
Examples
Examples of “surjection” in a Sentence
verb
British English
- The function surjects onto the entire codomain.
- We need to prove it surjects.
American English
- The map surjects onto the target set.
- To show it surjects, take an arbitrary element.
adverb
British English
- The function maps surjectively onto the space.
- It acts surjectively on the subgroup.
American English
- The relation holds surjectively.
- The projection operates surjectively.
adjective
British English
- This is a surjective mapping.
- The homomorphism is surjective.
American English
- That function is surjective.
- We require a surjective linear transformation.
Usage
Meaning in Context
Business
Not used.
Academic
Used exclusively in mathematics, computer science, and theoretical papers.
Everyday
Never used.
Technical
Core term in set theory, algebra, and functional programming type systems.
Vocabulary
Synonyms of “surjection”
Strong
Neutral
Weak
Vocabulary
Antonyms of “surjection”
Watch out
Common Mistakes When Using “surjection”
- Confusing surjection with injection or bijection.
- Using 'surjection' as a verb (e.g., 'It surjects' is rare and highly technical).
- Mispronouncing /sərˈdʒɛkʃən/ as /ˈsɜːrdʒɛkʃən/.
FAQ
Frequently Asked Questions
Yes, in standard mathematical terminology, 'surjection' and 'onto function' are synonyms.
Yes, such a function is called a bijection (one-to-one correspondence).
Extremely rarely. It may appear in highly theoretical computer science or logic, but almost never in everyday language.
A function that is not surjective (does not cover the entire codomain). There is no single-word antonym, but 'injection' (one-to-one function) is often contrasted with it.
A mathematical function where every element in the codomain is mapped to by at least one element in the domain.
Surjection is usually technical/formal in register.
Surjection: in British English it is pronounced /səˈdʒɛkʃ(ə)n/, and in American English it is pronounced /sərˈdʒɛkʃən/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “None”
Learning
Memory Aids
Mnemonic
Think 'SUR' (like 'surface' or 'surround') + 'JECTION' (throwing). A surjection 'covers' the entire target surface by throwing elements onto every point.
Conceptual Metaphor
COMPLETE COVERAGE IS A SURJECTION (e.g., 'The newsletter surjects onto all department interests' – metaphorically meaning it covers every topic relevant to each department).
Practice
Quiz
Which of the following best describes a surjection?