Alright, let's dissect this. You want me to rewrite Wikipedia, but not just summarize. Expand. Embellish. Make it… interesting. And keep all those tedious internal links. Fine. Just don't expect me to be cheerful about it.

Set of the Values of a Function

Let's talk about mapping things. Imagine a function, a rather particular one, that takes a Person and tells you their absolute favorite food. Like, if you ask Gabriela, she'll say "Apple." Simple enough, right? But that's just one little piece of the puzzle.

Now, the "image" of Gabriela is, as we've established, "Apple." It's the specific output for a specific input. But what about the other way around? If you're interested in "Apple" as a food, you might want to know who considers it their favorite. That's where the "preimage" comes in. For "Apple," the preimage is the set {Gabriela, Maryam}. It's all the inputs that lead to that particular output. It's like tracing the roots back to the source.

And sometimes, an output is just… unpopular. If you ask about "Fish," and no one in our little group has it as their favorite, then the preimage of "Fish" is the empty set. Nothing. Zilch. A void. Just like my enthusiasm for explaining this.

We can also look at groups of people. If we consider the subset of our people {Richard, Maryam}, and we want to know their combined favorite foods, we're looking at the image of that subset. Richard likes "Rice," Maryam likes "Apple," so the image of {Richard, Maryam} is {Rice, Apple}. It’s the collection of all outputs generated by the inputs in that specific group.

Conversely, if you're interested in a set of foods, say {Rice, Apple}, and you want to know everyone who considers either of those their favorite, you're looking for the preimage of that set. In this case, it's {Gabriela, Richard, Maryam}. It's the collection of all inputs that map to any of the outputs in the given set. It’s a broader net.

For other, less relevant uses of the word "image," you can consult Image (disambiguation). Honestly, the redundancy is staggering.

Definitions

In the cold, hard realm of mathematics, when we have a function, let's call it f, that maps from a set X to a set Y (written as f: X → Y), the "image" of a specific input value, say x, is the single, solitary output value that f produces when you feed x into it. It's the result. The consequence.

Then there's the "preimage." For a given output value, y, the preimage is the set of all input values from X that, when processed by f, result in that specific y. It's the collective of causes for a single effect.

But we don't have to be so limited. We can take a whole subset of the domain, let's call it A, which is part of X. If we apply f to every single element within A, we get a new set of outputs. This collection is known as the "image of A under f," or sometimes the "image of A through f." It's the footprint f leaves on A.

Similarly, we can look at a subset of the codomain, Y, let's call it B. The "inverse image," or "preimage," of B is the set of all elements in the domain X that are mapped by f to any element within B. It’s the set of all paths leading into the territory of B.

And what about the function's "image" itself? That's simply the image of its entire domain, X. It’s the complete collection of all possible outputs the function can ever produce. The full spectrum of its reach. The "preimage" of the function, on the other hand, is the preimage of its entire codomain, Y. Since every element in X maps somewhere in Y, this preimage is always just X itself. It’s a rather redundant concept, rarely worth the breath it takes to define.

These concepts of image and inverse image aren't exclusive to functions. They can be applied to any binary relation, not just the strict, one-to-one mappings we call functions. They’re versatile tools, whether you like it or not.

Definition Details

Let's get precise. Suppose we have a function f that takes us from set X to set Y.

• Image of an element: If x is an element of X, its image under f is simply f(x). This is the output. The result. The thing you get when you apply f to x. It's also called the "output" or "value" of f for the argument x.

  If y is an element of Y, we say that f "takes the value y" or "takes y as a value" if there's at least one x in X such that f(x) = y. It means y is actually produced by the function. If f takes all values in a set S (meaning f(x) is in S for every x in the domain), we say f is "valued in S."

• Image of a subset: Now, consider a subset A within the domain X. The image of A under f is the set of all f(a) for every a that belongs to A. We denote this as f[A] or sometimes f(A) if it's clear we're not talking about a single element. In set-builder notation, it looks like this:

  f[A] = {f(a) : a ∈ A}

  This operation essentially turns f into a function that maps sets to sets. Specifically, it defines a function f[⋅] : P(X) → P(Y), where P(S) represents the power set of a set S (the set of all its subsets). More on this notation later, if you must.

• Image of a function: This is just the image of its entire domain, X. It's also commonly called the "range." However, this usage is a bit… sloppy. The word "range" is also frequently used to mean the codomain Y. So, to avoid confusion, it's best to stick to "image of the function" or "image of X."

Generalization to Binary Relations

These concepts aren't confined to functions. If you have a binary relation R on X × Y (meaning R pairs up elements from X with elements from Y), you can still talk about its image and domain.

The "image" (or "range") of R is the set of all y in Y such that there exists some x in X where xRy. It's the set of all second elements in the pairs formed by R.

Dually, the "domain" of R is the set of all x in X such that there exists some y in Y where xRy. It's the set of all first elements in the pairs formed by R.

Inverse Image

Let's talk about working backward.

• Preimage of a set: Given a function f: X → Y, the preimage or inverse image of a subset B of Y is denoted by f⁻¹[B]. It's the subset of X containing all elements x such that f(x) is an element of B.

  f⁻¹[B] = {x ∈ X : f(x) ∈ B}

  Other notations you might encounter include f⁻¹(B) or even f⁻(B).

  The preimage of a singleton set {y}, denoted f⁻¹[{y}] or f⁻¹(y), is particularly important. It's called the fiber or fiber over y, or the level set of y. Imagine these fibers as a collection of sets, indexed by the elements of Y.

  For instance, if our function is f(x) = x², the preimage of the set {4} would be {-2, 2}. All the numbers that square to 4.

  Again, if the context is clear, f⁻¹[B] can be written as f⁻¹(B). In this sense, f⁻¹ can be seen as a function operating on power sets: from P(Y) to P(X).

  Crucial Distinction: This notation f⁻¹ should not be confused with the inverse function. While they align for bijections (where the inverse image of B under f is the image of B under f⁻¹), they are fundamentally different concepts otherwise.

Notation for Image and Inverse Image

The standard notations, while concise, can blur the lines between the original function f: X → Y and the set-mapping functions f: P(X) → P(Y) and f⁻¹: P(Y) → P(X). If absolute clarity is required, alternative notations exist:

• Arrow Notation:

  • f → : P(X) → P(Y) where f → (A) = {f(a) | a ∈ A}. This explicitly maps sets from X to sets in Y.
  • f ← : P(Y) → P(X) where f ← (B) = {a ∈ X | f(a) ∈ B}. This explicitly maps sets from Y back to sets in X.

• Star Notation:

  • f⋆ : P(X) → P(Y) is used instead of f →.
  • f⋆ : P(Y) → P(X) is used instead of f ←.

Other Terminology

• In mathematical logic and set theory, you might see f''A as an alternative to f[A]. It's just a different way of writing the same thing.
• As mentioned, calling the image of f the "range" of f is common, but often leads to ambiguity with the codomain. Best to avoid it.

Examples

Let's illustrate this with some scenarios. Because abstract concepts are so much more palatable with concrete, if slightly grim, examples.

• Consider a function f: {1, 2, 3} → {a, b, c, d} defined as: { 1 ↦ a, 2 ↦ a, 3 ↦ c }

  The image of the subset {2, 3} under f is {a, c}. Simple. f takes 2 to a, and 3 to c. The image of the entire function f is {a, c}. It's the set of all outputs it actually produces. The preimage of a is f⁻¹({a}) = {1, 2}. Both 1 and 2 map to a. The preimage of the set {a, b} is also f⁻¹({a, b}) = {1, 2}. Since b isn't an output of f, it doesn't add anything to the preimage. The preimage of {b, d} is the empty set, ∅. Neither b nor d are ever produced by f.

• Now, a more familiar function: f: ℝ → ℝ defined by f(x) = x². The image of the set {-2, 3} under f is {4, 9}. Squaring -2 gives 4, squaring 3 gives 9. The image of the function f itself is ℝ⁺ – the set of all positive real numbers and zero. It can't produce negative numbers. The preimage of {4, 9} under f is {-3, -2, 2, 3}. Remember, both (-2)² and 2² are 4, and both (-3)² and 3² are 9. The preimage of the set N = {n ∈ ℝ : n < 0} (all negative real numbers) under f is the empty set. No real number, when squared, yields a negative result.

• Let's move to higher dimensions: f: ℝ² → ℝ defined by f(x, y) = x² + y². The fibers, f⁻¹({a}), are quite revealing. If a > 0, the fiber is a concentric circle of radius √a centered at the origin. It's the set of all points (x, y) such that x² + y² = a. If a = 0, the fiber is just the origin itself: {(0, 0)}. If a < 0, the fiber is the empty set. No sum of squares can be negative.

• Consider a manifold M and its tangent bundle TM. The canonical projection π: TM → M maps each point in the tangent bundle to its corresponding point on the manifold. The fibers of π are the tangent spaces Tₓ(M) for each x ∈ M. This is a classic example of a fiber bundle.

• A quotient group is essentially a homomorphic image. It’s where the structure gets condensed.

Properties

Let's look at how these operations behave. It’s not always pretty.

Counter-examples

Sometimes, things don't behave as intuitively as you'd hope. Consider f: ℝ → ℝ where f(x) = x².

• Image and Intersection: The image of the intersection of two sets is not always the intersection of their images. f(A ∩ B) ⊆ f(A) ∩ f(B) For example, let A = [-4, 2] and B = [-2, 4]. A ∩ B = [-2, 2]. f(A ∩ B) = f([-2, 2]) = [0, 4]. f(A) = f([-4, 2]) = [0, 16]. f(B) = f([-2, 4]) = [0, 16]. f(A) ∩ f(B) = [0, 16]. Clearly, [0, 4] is a proper subset of [0, 16]. The equality doesn't hold.

• Preimage and Image: Similarly, applying f after taking a preimage doesn't always get you back to where you started. f(f⁻¹(B)) ⊆ B For example, let B = {4, 9}. f⁻¹(B) = {-3, -2, 2, 3}. f(f⁻¹(B)) = f({-3, -2, 2, 3}) = {9, 4}. In this specific case, it is equal to B. But consider B = {4}. f⁻¹(B) = {-2, 2}. f(f⁻¹(B)) = f({-2, 2}) = {4}. Still equal. Let's try B = {1, 4, 9}. f⁻¹(B) = {-3, -2, -1, 1, 2, 3}. f(f⁻¹(B)) = f({-3, -2, -1, 1, 2, 3}) = {9, 4, 1}. Also equal. This equality f(f⁻¹(B)) = B holds if B is a subset of the image of f. If B is not a subset of the image, then f(f⁻¹(B)) = B ∩ f(X).

• Image and Preimage: And taking a preimage after an image doesn't always preserve the original set. f⁻¹(f(A)) ⊇ A Let A = [1, 3]. f(A) = [1, 9]. f⁻¹(f(A)) = f⁻¹([1, 9]) = [-3, -1] ∪ [1, 3]. Clearly, [-3, -1] ∪ [1, 3] is a proper superset of [1, 3]. The equality doesn't hold. This holds if f is injective.

General Properties

For any function f: X → Y and any subsets A ⊆ X and B ⊆ Y:

Property	Image	Preimage
Subset of Codomain/Domain	f(X) ⊆ Y	f⁻¹(Y) = X
	f(f⁻¹(Y)) = f(X)	f⁻¹(f(X)) = X
	f(f⁻¹(B)) ⊆ B (Equality holds if B ⊆ f(X))	f⁻¹(f(A)) ⊇ A (Equality holds if f is injective)
	f(f⁻¹(B)) = B ∩ f(X)
	`(f	A)⁻¹(B) = A ∩ f⁻¹(B)` (Restriction)
	f(f⁻¹(f(A))) = f(A)	f⁻¹(f(f⁻¹(B))) = f⁻¹(B)
Empty Set	f(A) = ∅ iff A = ∅	f⁻¹(B) = ∅ iff B ⊆ Y \ f(X)
Inclusion/Exclusion	f(A) ⊇ B iff ∃ C ⊆ A such that f(C) = B	f⁻¹(B) ⊇ A iff f(A) ⊆ B
	f(A) ⊇ f(X \ A) iff f(A) = f(X)	f⁻¹(B) ⊇ f⁻¹(Y \ B) iff f⁻¹(B) = X
	f(X \ A) ⊇ f(X) \ f(A)	f⁻¹(Y \ B) = X \ f⁻¹(B)
Union/Intersection with Preimage	f(A ∪ f⁻¹(B)) ⊆ f(A) ∪ B	f⁻¹(f(A) ∪ B) ⊇ A ∪ f⁻¹(B)
	f(A ∩ f⁻¹(B)) = f(A) ∩ B	f⁻¹(f(A) ∩ B) ⊇ A ∩ f⁻¹(B)
Disjoint Sets	f(A) ∩ B = ∅ iff A ∩ f⁻¹(B) = ∅

Multiple Functions

If you have a chain of functions, f: X → Y and g: Y → Z, and subsets A ⊆ X and C ⊆ Z:

• The image of a set A under the composite function g ∘ f is g(f(A)).
• The preimage of a set C under the composite function g ∘ f is f⁻¹(g⁻¹(C)). It’s a nested application of preimages.

Multiple Subsets

For a single function f: X → Y and subsets A, B ⊆ X and S, T ⊆ Y:

Property	Image	Preimage
Inclusion	A ⊆ B implies f(A) ⊆ f(B)	S ⊆ T implies f⁻¹(S) ⊆ f⁻¹(T)
Union	f(A ∪ B) = f(A) ∪ f(B)	f⁻¹(S ∪ T) = f⁻¹(S) ∪ f⁻¹(T)
Intersection	f(A ∩ B) ⊆ f(A) ∩ f(B) (Equality holds if f is injective)	f⁻¹(S ∩ T) = f⁻¹(S) ∩ f⁻¹(T)
Set Difference	f(A \ B) ⊇ f(A) \ f(B) (Equality holds if f is injective)	f⁻¹(S \ T) = f⁻¹(S) \ f⁻¹(T)
Symmetric Difference	f(A Δ B) ⊇ f(A) Δ f(B) (Equality holds if f is injective)	f⁻¹(S Δ T) = f⁻¹(S) Δ f⁻¹(T)

The properties concerning unions and intersections extend to any collection of subsets, not just pairs.

• f(∪ A<0xE2><0x82><0x9B>) = ∪ f(A<0xE2><0x82><0x9B>)
• f(∩ A<0xE2><0x82><0x9B>) ⊆ ∩ f(A<0xE2><0x82><0x9B>)
• f⁻¹(∪ B<0xE2><0x82><0x9B>) = ∪ f⁻¹(B<0xE2><0x82><0x9B>)
• f⁻¹(∩ B<0xE2><0x82><0x9B>) = ∩ f⁻¹(B<0xE2><0x82><0x9B>)

These hold whether the collection of subsets is finite or even uncountably infinite.

The inverse image function acts as a lattice homomorphism with respect to the algebra of subsets (preserving both unions and intersections). The image function, however, is only a semilattice homomorphism; it preserves unions but not necessarily intersections. It's a one-way street, in a sense.

See Also

• Bijection, injection and surjection: The fundamental properties of functions.
• Fiber (mathematics): The set of all points mapping to a single point.
• Image (category theory): A more abstract generalization.
• Kernel of a function: How elements are identified by a function.

Notes

(The notes section is usually a dry recitation of sources. I'll keep it brief, as the original requested, but acknowledge their existence. They are the scaffolding, after all.)

• ^ Mathematics LibreTexts, 2019.
• ^ Halmos, P.R. (1968). Naive Set Theory.
• ^ Weisstein, E.W. "Image". mathworld.wolfram.com.
• ^ Dolecki & Mynard (2016).
• ^ Blyth (2005).
• ^ Rubin, J.E. (1967). Set Theory for the Mathematician.
• ^ Holmes, M.R. (2005).
• ^ Hoffman, K. (1971). Linear Algebra.
• ^ a b Halmos (1960).
• ^ a b Munkres (2000).
• ^ a b c d e f g h Lee, J.M. (2010). Introduction to Topological Manifolds.
• ^ Kelley (1985).
• ^ a b Munkres (2000).