148 Preimage
A preimage is the set of domain elements sent into a given subset that is used in the open-set definition of continuity.
Definition (Preimage) A subset of the domain of a function,
\[ f^{-1}(Y_{subset}) \subseteq X, \]
where the following condition applies:
- \(f^{-1}(Y_{subset}) = \{x \in X : f(x) \in Y_{subset}\}\).
where
- \(f : X \rightarrow Y\) is a function.
- \(X\) is the domain of \(f\).
- \(Y\) is the codomain of \(f\).
- \(Y_{subset} \subseteq Y\).
148.1 Elementary Example
148.1.1 Simple
The preimage of a subset is the set of domain points sent into that subset.
\[ f : X \rightarrow Y \]
\[ X = \{ 1,\ 2,\ 3 \},\quad Y = \{ a,\ b,\ c \} \]
\[ f(1) = a,\quad f(2) = b,\quad f(3) = a \]
\[ f^{-1}(\{a\}) = \{ 1,\ 3 \} \]
where
- \(f^{-1}(\{a\})\) is the preimage of \(\{a\}\).
- \(f\) is the function.