Image (mathematics)

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For the cryptographic attack on hash functions, see preimage attack.

In mathematics, image is a part of the set theoretic notion of function.

1 Definition
2 Examples
3 Consequences
4 See also

[edit] Definition

Let X and Y be sets, f be the function f : X → Y, and x be some member of X. Then the image of x under f, denoted f(x), is the unique member y of Y that f associates with x. The image of a function f is denoted im(f) and is the range of f, or more precisely, the image of its domain.

The image of a subset A ⊆ X under f is the subset of Y defined by

f[A] = {y ∈ Y | y = f(x) for some x ∈ A}.

When there is no risk of confusion, f[A] is sometimes simply written f(A). An alternative notation for f[A], common in the older literature on mathematical logic and still preferred by some set theorists, is f "A.

Given this definition, the image of f becomes a function whose domain is the power set of X (the set of all subsets of X), and whose codomain is the power set of Y. The same notation can denote either the function f or its image. This convention is a common one; the intended meaning must be inferred from the context.

The preimage or inverse image of a set B ⊆ Y under f is the subset of X defined by

f⁻¹[B] = {x ∈ X | f(x) ∈ B}.

The inverse image of a singleton, f⁻¹[{y}], is a fiber (also spelled fibre) or a level set.

Again, if there is no risk of confusion, we may denote f⁻¹[B] by f⁻¹(B), and think of f⁻¹ as a function from the power set of Y to the power set of X. The notation f⁻¹ should not be confused with that for inverse function. The two coincide only if f is a bijection.

f can also be seen as a family of sets indexed by Y, which leads to the notion of a fibred category.

[edit] Examples

1. f: {1,2,3} → {a,b,c,d} defined by $f(x)=\left\{\begin{matrix} a, & \mbox{if }x=1 \\ d, & \mbox{if }x=2 \\ c, & \mbox{if }x=3. \end{matrix}\right.$

The image of {2,3} under f is f({2,3}) = {d,c}, and the range of f is {a,d,c}. The preimage of {a,c} is f⁻¹({a,c}) = {1,3}.

2. f: R → R defined by f(x) = x².

The image of {-2,3} under f is f({-2,3}) = {4,9}, and the range of f is R⁺. The preimage of {4,9} under f is f⁻¹({4,9}) = {-3,-2,2,3}.

3. f: R² → R defined by f(x, y) = x² + y².

The fibres f⁻¹({a}) are concentric circles about the origin, the origin, and the empty set, depending on whether a>0, a=0, or a<0, respectively.

4. If M is a manifold and π :TM→M is the canonical projection from the tangent bundle TM to M, then the fibres of π are the tangent spaces T_x(M) for x∈M. This is also an example of a fiber bundle.

[edit] Consequences

Given a function f : X → Y, for all subsets A, A₁, and A₂ of X and all subsets B, B₁, and B₂ of Y we have:

f(A₁ ∪ A₂) = f(A₁) ∪ f(A₂)
f(A₁ ∩ A₂) ⊆ f(A₁) ∩ f(A₂)
f⁻¹(B₁ ∪ B₂) = f⁻¹(B₁) ∪ f⁻¹(B₂)
f⁻¹(B₁ ∩ B₂) = f⁻¹(B₁) ∩ f⁻¹(B₂)
f(f⁻¹(B)) ⊆ B
f⁻¹(f(A)) ⊇ A
A₁ ⊆ A₂ → f(A₁) ⊆ f(A₂)
B₁ ⊆ B₂ → f⁻¹(B₁) ⊆ f⁻¹(B₂)
f⁻¹(B^C) = (f⁻¹(B))^C
(f |_A)⁻¹(B) = A ∩ f⁻¹(B).

The results relating images and preimages to the (Boolean) algebra of intersection and union work for any collection of subsets, not just for pairs of subsets.

With respect to the algebra of subsets, by the above we see that the inverse image function is a lattice homomorphism while the image function is just a semilattice homomorphism (it does not always preserve intersections).