Image and preimage

Definition: Image, preimage

Given a function $f:X\to Y$, we can produce subsets of $X$ and $Y$:

• If $S\subset X$, the (forward) image of $S$ under the function $f$ is the set of points in $Y$ which are mapped to by some point(s) in $S$.

$$f(S)=\{y\in Y\mid y=f(s)\text{ for some }s\in S\}$$

• If $T\subset Y$, the preimage or reverse image of $T$ under the function $f$ is the set of points in $X$ which map to some point in $T$.

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

Proposition: Relations between image and preimage sets

Let $f:X\to Y$, $A_1,A_2\subseteq X$, and $B_1,B_2\subseteq Y$. Then:

• Subsets. If $A_1\subseteq A_2$, then $f(A_1)\subseteq f(A_2)$; similarly, if $B_1\subseteq B_2$ then $f^{-1}(B_1)\subseteq f^{-1}(B_2)$
• Unions. $f(A_1\cup A_2)=f(A_1)\cup f(A_2)$; similar equality holds for the preimage and $B_1\cup B_2$
• Intersections. $f(A_1\cap A_2)\subseteq f(A_1)\cap f(A_2)$ is an inclusion, while $f^{-1}(B_1\cap B_2)=f^{-1}(B_1)\cap f^{-1}(B_2)$ is an equality

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Review §

• Prove the relations between image and preimage sets.