inverse image
RareAcademic/Technical
Definition
Meaning
The set of all elements in a function's domain that map to a given subset of its codomain.
The preimage of a subset under a function; often used to analyse the relationship between elements in two sets via a mapping.
Linguistics
Semantic Notes
Primarily a term from mathematics (set theory, topology, algebra) with occasional metaphorical use in technical or philosophical writing to describe a conceptual reverse-mapping process.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or spelling differences. Usage is identical across both standard varieties.
Connotations
Purely technical, neutral connotations.
Frequency
Exclusively found in mathematical and related technical contexts; identical low frequency in both varieties.
Vocabulary
Collocations
Grammar
Valency Patterns
[noun phrase] + of + [noun phrase] (e.g., the inverse image of B)[determiner] + inverse image + under + [function] (e.g., the inverse image under f)Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Usage
Context Usage
Business
Virtually never used.
Academic
Core term in mathematics lectures and textbooks.
Everyday
Not used.
Technical
Essential in pure and applied mathematics, computer science (especially functional programming), and engineering contexts involving mappings.
Examples
By Part of Speech
verb
British English
- To find the inverse image, one must consider all elements mapping to the given set.
American English
- We need to inverse-image the target subset to understand the domain's structure.
adjective
British English
- The inverse-image set was surprisingly large.
American English
- We studied the inverse-image properties of continuous functions.
Examples
By CEFR Level
- The inverse image of a single point under a many-to-one function can contain multiple elements.
- In our graph, the inverse image of the interval [0,1] is the entire left-hand region.
- The continuity of the function guarantees that the inverse image of an open set is itself open.
- By analysing the inverse image of the critical subspace, we derived the structure of the solution set.
Learning
Memory Aids
Mnemonic
Think: INVERSE means opposite direction. IMAGE is what comes out. So INVERSE IMAGE is going backwards from the output to find what inputs produced it.
Conceptual Metaphor
TRACING BACKWARDS TO FIND ORIGINS.
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Avoid direct translation that might imply a "reversed picture". The established Russian term is "прообраз". Do not confuse with "обратное изображение", which is incorrect.
Common Mistakes
- Using it to mean 'mirror image' or 'opposite picture'.
- Confusing 'inverse image' (preimage) with 'inverse function'. The inverse image is a set, not necessarily a function.
Practice
Quiz
What is the inverse image of a set B under a function f?
FAQ
Frequently Asked Questions
Yes, they are completely synonymous terms in mathematics.
No. The inverse image (or preimage) is defined for any function, regardless of whether it has an inverse function.
Yes. If no element in the domain maps into the given subset of the codomain, the inverse image is the empty set.
Primarily in pure mathematics (set theory, topology, analysis), but also in applied mathematics, theoretical computer science, mathematical logic, and some branches of physics and engineering.