Preimage vs. Image — What's the Difference?
Edited by Tayyaba Rehman — By Urooj Arif — Updated on March 26, 2024
Preimage involves the original set before a function's application, whereas the image is the resultant set after the function acts on the preimage.
Difference Between Preimage and Image
Table of Contents
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Key Differences
Preimage refers to the set of all original inputs in a function that maps to a particular output or set of outputs. On the other hand, the image is the set of all outputs resulting from applying the function to its preimage. This fundamental difference highlights the directional flow of a function from input to output.
Preimages are crucial for understanding functions' reversibility, as they identify which inputs lead to specific outputs. Whereas, images help in determining the function's range, showcasing the actual outputs that the function can produce.
In the context of mathematics, particularly in functions and mappings, the preimage can sometimes be a single value or a set of values that map to a specific image. Conversely, an image is always a specific outcome of the function, which could result from one or multiple elements of the preimage.
Preimage and image concepts are applied beyond pure mathematics, including in computer science for hashing and encryption, where understanding the relationship between original data (preimage) and processed data (image) is essential. While the preimage might represent original, unencrypted data, the image relates to the encrypted or hashed result.
The exploration of preimages and images is vital for comprehending the behavior of functions, including their injectivity, surjectivity, and bijectivity. While the concept of a preimage focuses on the breadth of inputs that lead to certain outputs, the image illustrates the function's effectiveness in covering its possible output domain.
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Comparison Chart
Definition
The original set of inputs before a function's application.
The resultant set of outputs after a function's application.
Role in Function
Identifies possible inputs for a given output.
Represents actual outputs for given inputs.
Mathematical Context
Often a set of values mapping to a specific output.
Specific outcomes of the function.
Relevance
Important for assessing function's reversibility.
Crucial for determining function's range.
Applications
Used in cryptography for original data.
Applied in hashing for processed data.
Compare with Definitions
Preimage
The set of all original inputs that map to a particular output in a function.
In a square function, the preimage of 4 could be -2 or 2.
Image
The set of all outputs produced by applying a function to its preimage.
The image of 3 under a doubling function is 6.
Preimage
Represents original, unencrypted data.
The preimage of a hash function is the original data before hashing.
Image
Demonstrates the range of a function.
The image set of a function defines its output possibilities.
Preimage
Assists in exploring a function's injectivity.
A function with unique preimages for each image is injective.
Image
Refers to encrypted or hashed data.
The image of a plaintext under an encryption algorithm is ciphertext.
Preimage
Used for understanding inverse functions.
The preimage of a value under an inverse function is its original input value.
Image
Used to study a function's surjectivity.
A function whose image covers the entire output domain is surjective.
Preimage
Helps identify inputs that lead to specific outputs.
Finding the preimage under a square root function involves identifying squares.
Image
Important for mapping under transformation.
The image of a curve under a transformation shows its new position.
Preimage
The set of arguments of a function corresponding to a particular subset of the range.
Image
An image (from Latin: imago) is an artifact that depicts visual perception, such as a photograph or other two-dimensional picture, that resembles a subject—usually a physical object—and thus provides a depiction of it. In the context of signal processing, an image is a distributed amplitude of color(s).
Preimage
(mathematics) For a given function, the set of all elements of the domain that are mapped into a given subset of the codomain; (formally) given a function ƒ : X → Y and a subset B ⊆ Y, the set ƒ−1(B) = {x ∈ X : ƒ(x) ∈ B}.
The preimage of under the function is the set .
Image
A representation of the external form of a person or thing in art
Her work juxtaposed images from serious and popular art
Image
The general impression that a person, organization, or product presents to the public
She strives to project an image of youth
Image
A simile or metaphor
He uses the image of a hole to describe emotional emptiness
Image
Make a representation of the external form of
Artworks which imaged women's bodies
Image
A representation of the form of a person or object, such as a painting or photograph.
Image
A sculptured likeness.
Image
(Physics) An optically formed duplicate, counterpart, or other representative reproduction of an object, especially an optical reproduction formed by a lens or mirror.
Image
One that closely or exactly resembles another
He is the image of his uncle.
Image
Likeness; semblance
Genesis says that man was made in the image of God.
Image
The opinion or concept of something that is held by the public
The public's image of business leaders as greedy.
Image
The concept or character projected to the public, as by a person or institution, especially as interpreted by the mass media
An actor who tried to convey an image of refined beauty.
Image
A typical example or embodiment
That child is the image of good health.
Image
A mental picture of something not real or present
Our image of the cottage did not conform with reality.
Image
A vivid description or representation in words, especially a metaphor or simile
The poem uses the image of a barren tree to convey feelings of desolation.
Image
(Mathematics) A set of values of a function corresponding to a particular subset of a domain.
Image
(Computers) An exact replica of the contents of a storage device, such as a hard disk, stored on a second storage device, such as a network server.
Image
(Obsolete) An apparition.
Image
To make or produce a likeness of
Imaged the poet in bronze.
Image
To mirror or reflect
A statue imaged in the water.
Image
To make a visual representation of (an object) using remote scanning or technology such as magnetic resonance imaging
Imaged the diseased kidneys.
Imaged the surface of Mars.
Image
To symbolize or typify
A kneeling woman imaging the nation's grief.
Image
To picture mentally; imagine or visualize
Imaged each dive before doing it.
Image
To describe, especially so vividly as to evoke a mental picture
The passage images what it's like to grow up poor.
Image
To print (a file) using a laser printer, imagesetter, direct-to-plate press, or similar device.
Image
To transmit (an exact replica of the contents of a storage device) to another storage device
Imaged the hard drive to the server.
Image
An optical or other representation of a real object; a graphic; a picture.
The Bible forbids the worship of graven images.
Image
A mental picture of something not real or not present.
Image
A statue or idol.
Image
(computing) A file that contains all information needed to produce a live working copy. (See disk image and image copy.)
Most game console emulators do not come with any ROM images for copyright reasons.
Image
A characteristic of a person, group or company etc., style, manner of dress, how one is or wishes to be perceived by others.
Image
(mathematics) What a function maps to.
The number 6 is the image of 3 under f that is defined as f(x) = 2x.
Image
(mathematics) The subset of a codomain comprising those elements that are images of something.
The image of this step function is the set of integers.
Image
(radio) A form of interference: a weaker "copy" of a strong signal that occurs at a different frequency.
Image
(obsolete) Show; appearance; cast.
Image
(transitive) To represent by an image or symbol; to portray.
Image
(transitive) To reflect, mirror.
Image
(transitive) To create an image of.
Image
To create a complete backup copy of a file system or other entity.
Image
An imitation, representation, or similitude of any person, thing, or act, sculptured, drawn, painted, or otherwise made perceptible to the sight; a visible presentation; a copy; a likeness; an effigy; a picture; a semblance.
Even like a stony image, cold and numb.
Whose is this image and superscription?
This play is the image of a murder done in Vienna.
And God created man in his own image.
Image
Hence: The likeness of anything to which worship is paid; an idol.
Thou shalt not make unto thee any graven image, . . . thou shalt not bow down thyself to them.
Image
Show; appearance; cast.
The face of things a frightful image bears.
Image
A representation of anything to the mind; a picture drawn by the fancy; a conception; an idea.
Can we conceiveImage of aught delightful, soft, or great?
Image
A picture, example, or illustration, often taken from sensible objects, and used to illustrate a subject; usually, an extended metaphor.
Image
The figure or picture of any object formed at the focus of a lens or mirror, by rays of light from the several points of the object symmetrically refracted or reflected to corresponding points in such focus; this may be received on a screen, a photographic plate, or the retina of the eye, and viewed directly by the eye, or with an eyeglass, as in the telescope and microscope; the likeness of an object formed by reflection; as, to see one's image in a mirror.
Image
To represent or form an image of; as, the still lake imaged the shore; the mirror imaged her figure.
Image
To represent to the mental vision; to form a likeness of by the fancy or recollection; to imagine.
Condemn'd whole years in absence to deplore,And image charms he must behold no more.
Image
An iconic mental representation;
Her imagination forced images upon her too awful to contemplate
Image
A visual representation (of an object or scene or person or abstraction) produced on a surface;
They showed us the pictures of their wedding
A movie is a series of images projected so rapidly that the eye integrates them
Image
(Jungian psychology) a personal facade that one presents to the world;
A public image is as fragile as Humpty Dumpty
Image
A standard or typical example;
He is the prototype of good breeding
He provided America with an image of the good father
Image
Language used in a figurative or nonliteral sense
Image
Someone who closely resembles a famous person (especially an actor);
He could be Gingrich's double
She's the very image of her mother
Image
A representation of a person (especially in the form of sculpture);
The coin bears an effigy of Lincoln
The emperor's tomb had his image carved in stone
Image
Imagine; conceive of; see in one's mind;
I can't see him on horseback!
I can see what will happen
I can see a risk in this strategy
Common Curiosities
What is a preimage?
A preimage is the set of all inputs that map to a particular output in a function.
What does the image of a function mean?
The image of a function refers to the set of all outputs it can produce from its inputs.
How does the concept of preimage apply in cryptography?
In cryptography, the preimage refers to the original data before it undergoes encryption or hashing.
What role does the image play in understanding a function's range?
The image set of a function helps in identifying its range, showing all possible outputs.
Can a single output have multiple preimages?
Yes, a single output can have multiple preimages, especially in non-injective functions.
Can the preimage be the same as the image?
In certain functions, like the identity function, the preimage and image can be the same.
Can there be an image without a preimage in a function?
No, in a function, every image must have at least one preimage, by definition.
How are preimage and image used in function analysis?
They are used to assess functions' injectivity, surjectivity, and bijectivity.
Is the concept of image relevant outside mathematics?
Yes, it's also relevant in fields like computer science, especially in hashing and encryption.
What determines a function's surjectivity?
A function is surjective if its image covers the entire output domain.
What is the difference between preimage and inverse in mathematical functions?
Preimage refers to inputs that map to a given output, while an inverse function reverses the direction of mapping.
What is the importance of finding a function's preimage?
Finding a function's preimage helps in understanding which inputs lead to specific outputs.
How do preimage and image contribute to cryptography security?
They are crucial in analyzing the security of cryptographic functions, ensuring data integrity and confidentiality.
Why is the concept of a preimage significant in function reversibility?
It indicates whether a function can be reversed to obtain original inputs, highlighting its injectivity or bijectivity.
In what way is the concept of an image important in calculus?
It's important for mapping and transformation, showing how functions modify their inputs.
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Written by
Urooj ArifUrooj is a skilled content writer at Ask Difference, known for her exceptional ability to simplify complex topics into engaging and informative content. With a passion for research and a flair for clear, concise writing, she consistently delivers articles that resonate with our diverse audience.
Edited by
Tayyaba RehmanTayyaba Rehman is a distinguished writer, currently serving as a primary contributor to askdifference.com. As a researcher in semantics and etymology, Tayyaba's passion for the complexity of languages and their distinctions has found a perfect home on the platform. Tayyaba delves into the intricacies of language, distinguishing between commonly confused words and phrases, thereby providing clarity for readers worldwide.
