Hello, in this question:

Let f be a function from set A to set B for a set, X subset B define f-1(X) = |x element A: f(x) subset X|. Then which of the following is necessarily true for a subset U of X?

Im confused about what it is asking and the notation. For example, i dont understand what f would be in this case.

As the options are not available, question seems incomplete.

Though I can help you with the understanding of the question with an example:

f: A→B such that y€B. Then, f^-1{y}= { x€ A : f(x) = y}.

This means thatf-1{y} is the set of pre-images of y
eg. Question. Let f: R→R be a function given by f(x)=x²+1. Find f^-1{26}
Sol.
Letf-1{26}=x Then f(x)=26

x2+1= 26
x= +or- 5
f-1{26}={-5,5}