Image (mathematics) Guide, Meaning , Facts, Information and Description

The image of an element x∈X under the function f:X→Y, denoted by f(x), is the unique y∈Y that is associated with x.

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

f(A) = {f(x) | x in A}

Notice that the range of f is the image f(X) of its domain X.

Note that with this definition, the direct image f becomes a function whose domain is the set of all subsets of X (also known as the power set of X) and whose codomain is the power set of Y. Note that the same notation is used for the original function f and its direct image. This is a common convention; the intended usage must be inferred by context.

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

f ⁻¹(B) = {x in X | f(x) ∈B}

Note that with this definiton, f ⁻¹ becomes a function whose domain is the power set of Y and whose codomain is the power set of X.

Example

1. f: {1,2,3} → {a,b,c,d} defined by

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

2. f: R → R defined by f(x)=x².

In this example, the image of [-2,3] under f is f([-2,3])=[0,9] and the range of f is the set of nonnegative real numbers. The preimage of [-2,3] under f is f ⁻¹([0,9])=[-3,3].

Consequences

Some consequences that follow immediately from these definitions are:

• 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

These are valid for arbitrary subsets A, A₁ and A₂ of the domain and arbitrary subsets B, B₁ and B₂ of the codomain. The results relating images and preimages to the algebra of intersection and union work for any collections of subsets, not just for pairs of subsets.

See also:

• Image (category theory)
• Kernel of a function

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