To save me some time writing everything out in latex, I'm adding a picture of the question and Ill try to explain what I understand for the problem. Just a heads up, I'm really not sure how to do this at all :/
So the definition is saying for a function G, A implies B and elements of subset C are also elements of subset B. The inverse function (given) is elements a for which g(a) is in the set, C?
Other than that, I'm fairly lost on how to do any of the a,b,c parts. I'm not asking for straight up answers for all three, but to actually understand/make sense as to what the question is asking/how I can find the solutions.
solutions:
a.) f$^-1$({a,-2,16}) = {1,-1,2,-2}
b.)
c.) [0,$\infty$) ?
Thanks!
First, the notation $g:A\longrightarrow B$ does not mean $A$ implies $B$ but "$g$ is a function with domain $A$ and codomain $B$", so that $g(x)$ only makes sense if $s\in A$, and in that case, $g(x)\in B$.
for instance, there is a function from the set of people to the set of integers which gives the age in years of a person. So you can write $age: \mathrm{People} \to \mathbb N$ and $age(5)$ doe not make sense, but $age(John Doe)$ makes sense and it has a value, say $45\in\mathbb N$.
Now the inverse image $g^{-1}$ for subsets is defined as follows: for a subset of the codomain $C\subset B$, it gives the subset $g^{-1}(C)\subset A$ which consists of the elements whose image is in $C$. For instance, $age^{-1}(\{13, ..., 19\})$ is the set of teenagers which is a subset of the set of people.
Concretely, how to answer the questions in the exercise ? For each value in the subset of $B$ given, look in $A$ for all the elements which are mapped to this value. Regrouping all those elements in a subset gives you the inverse image subset of $A$.