Definition 0.2.10 Let $A$ and $B$ be sets, and $f: A \rightarrow B$ be a function. Let $S \subseteq B$. Then the set $f^{-1}(S)=\{ x\in A : f(x) \in S\}$ is called the inverse image of the set $S$ under the function $f$.
Definition 0.2.14 Let $A$ and $B$ be sets, and $f: A \rightarrow B$ be a function. Let $T \subseteq A$. Then the set $f(T)=\{x\in B : x=f(t) \text{ for some }t \in T \}$
is called the image of the set $T$ under the function $f$.
How do these two definitions differ from traditional usage of $f$ and $f^{-1}$ as in a calculus course?
My attempt to make any sense of it:
They are similar in many ways. In the definitions, for a function $f$ that maps $A$ to $B$ is similar to plugging a value $x$ into $f(x)$ and getting an output $y$. But, in the definitions given above, $f$ is invertible if it is one-to-one and onto, but how does this differ from the traditional $f$ used in calculus? I cannot seem to find the ties or correlations.
Best Answer
It is common and convenient in many subjects to assume that some set $A$ that we are interested in is an anti-transitive set: That no member of $A$ is a subset of $A.\, $ E.g. we usually assume that a subset of $\Bbb R$ is never a member of $\Bbb R,$ so that if $f:\Bbb R \to \Bbb R$ and $T\subset \Bbb R$ then $f(T)=\{f(x):x\in T\}$ and $f^{-1}(T)=f^{-1}T=\{x:f(x)\in T\}$ are unambiguous definitions. And if $A,B$ are anti-transitive sets and $f:A\to B$ is one-to-one (injective) and if $b$ belongs to the (unambiguous) set $f(A)$, then defining $a=f^{-1}(b)\iff f(a)=b$ is also unambiguous. The anti-transitive assumption occurs, often tacitly, in many books and papers.
Difficulties arise, especially in Set Theory, when we cannot safely assume anti-transitivity. E.g. if $\emptyset$ and $\{\emptyset\}$ both belong to the domain of a function $f$, then it is unclear what $f(\{\emptyset \})$ "should" mean. In Set Theory a function $f:A\to B$ $is$ its graph, so $f$ is a certain type of subset of $A\times B.$ And in Set Theory it is preferable to write $b=f(a)$ to only mean that $(a,b)\in f.$ And then the notation $f''T$ (read $f$-double-prime-$T$) is used to denote $\{f(t): t\in T\cap dom(f)\},$ the image of $T$ under $f.$