Preimage of Function

up:: 020 MOC Mathematics

Given a function $f:X\to Y$ and a subset $B\subseteq Y$, we have that its preimage is the set of all points in $f$‘s domain mapping onto it.

Formally, it is

$$f^{-1}(B)=\{x\in X\mid f(x)\in B\}$$

When talking about points in the preimage of some subset,

$$x\in f^{-1}(B)⟺\exists y\in B\mid f(x)=y⟺f(x)\in B$$

Properties

• The preimage of the union is the union of the preimages
• The preimage of the intersection is the intersection of the preimages
• The preimage of the complement is the complement of the preimage