# Lecture 017

Range: can mean codomain or image (don't use the word range)

Image: Let A and B be sets and $f:A\rightarrow B$ be a function. Let $X \subseteq A$.

• Definition: image of X under f is $Im_f(X)=\{b\in B | (\exists a \in X)(f(a) = b)\}$

• When X = A, image of f is $Im_f(A) \subseteq B$

• $(\forall X\subseteq A)(Im_f(X) \subseteq B)$

• think of $Im_f: \mathcal{P}(A) \rightarrow \mathcal{P}(B)$

Preimage: Let A and B be sets and $f:A\rightarrow B$ be a function. Let $Y \subseteq B$.

• Definition: preimage of Y under f is $PreIm_f(Y)=\{a\in A | f(a) \in Y)\} \subseteq A$

• When Y = B, preimage of f is $PreIm_f(B) \subseteq A$

• think of $PreIm_f: \mathcal{P}(B) \rightarrow \mathcal{P}(A)$

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