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onto function

Very Low
UK/ˌɒn.tuː ˈfʌŋk.ʃən/US/ˌɑːn.tuː ˈfʌŋk.ʃən/

Technical

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Definition

Meaning

A function where every element of the function's codomain is the image of at least one element from its domain.

In mathematics, a surjective function; a mapping from a set X to a set Y such that for every y in Y there is at least one x in X with f(x) = y.

Linguistics

Semantic Notes

The term is primarily used in set theory, discrete mathematics, and formal logic. 'Onto' and 'surjective' are synonymous. It describes a complete covering of the target set.

Dialectal Variation

British vs American Usage

Differences

No significant differences in usage. The term is standard in mathematical English globally.

Connotations

Purely technical, no connotative differences.

Frequency

Equally low-frequency in both varieties, confined to academic/technical contexts.

Vocabulary

Collocations

strong
prove an onto functiondefine an onto functionsurjective and onto function
medium
map ontofunction from A onto Bis onto
weak
mathematical ontoonto mappingonto property

Grammar

Valency Patterns

[Noun Phrase] is an onto functionf: X → Y is ontoThe function maps A onto B

Vocabulary

Synonyms

Strong

surjection

Neutral

surjective function

Weak

onto mapping

Vocabulary

Antonyms

into functionnon-surjective functioninjective function (in the sense of not necessarily covering the codomain)

Usage

Context Usage

Academic

Used in pure and applied mathematics, computer science theory, and engineering courses dealing with discrete structures.

Everyday

Virtually never used.

Technical

The primary domain of use. Precise definition is crucial in proofs and algorithm design.

Examples

By Part of Speech

adjective

British English

  • The mapping was proven to be onto.
  • We require an onto homomorphism for this proof.

American English

  • The function is onto.
  • An onto linear transformation is necessary.

Examples

By CEFR Level

B2
  • The professor explained that not every function is an onto function.
  • A simple example of an onto function is f(x) = x from real numbers to real numbers.
C1
  • To solve the problem, we constructed an onto function from the set of vertices to the set of colours.
  • The theorem only holds if we can demonstrate the existence of an onto function between the two groups.

Learning

Memory Aids

Mnemonic

Think of the codomain as a target. An 'onto' function lands ON TO every single spot on the target.

Conceptual Metaphor

COMPLETE COVERAGE (like paint covering every part of a wall, or a postman delivering mail to every address in a town).

Watch out

Common Pitfalls

Translation Traps (for Russian speakers)

  • Avoid directly translating 'onto' as 'на' in isolation. The Russian equivalent is 'сюръективная функция'.
  • The phrase 'onto' here is a mathematical term of art, not the preposition.

Common Mistakes

  • Confusing 'onto function' with the preposition 'onto' (e.g., 'step onto').
  • Mistaking it for 'one-to-one function' (injective).
  • Using it in non-mathematical contexts.

Practice

Quiz

Fill in the gap
A function f: A → B is called an function if for every b in B, there exists an a in A such that f(a) = b.
Multiple Choice

Which of the following is synonymous with 'onto function'?

FAQ

Frequently Asked Questions

Yes, they are completely synonymous terms in mathematics.

Yes, such a function is called a 'bijective function' or a 'one-to-one correspondence'.

It is used almost exclusively in mathematics, computer science, and related technical fields.

A function that is not onto is called 'non-surjective' or an 'into function', meaning some elements of the codomain are not mapped to.