Image of Function

up:: 020 MOC Mathematics

Given a function $f:X\to Y$ and a subset $A\subset X$, we say the image of $A$ is the set of all points in $Y$ which are reached by $f$ from $A$.

Formally, it is

$$f(A)=\{y\in Y\mid \exists a\in A:f(a)=y\}$$

Thus, when talking about points in the image of some subset,

$$y\in f(A)⟺\exists x\in A\mid f(x)=y$$

Note that Every function is surjective with respect to its image, since all points in the image are, per definition, reached by some point in its domain.

Properties

• The image of the union is the union of the images
• The image of the intersection is contained in the intersection of the images
  • The injective image of intersections is the intersection of images
• The image of the complement contains the complement of the images
  • The injective image of complements is the complement of images
• The image of the preimage of a set is contained in the set
• The preimage of the image of a set contains the set