# Proof about pre-images of functions

• I
PhysicsRock
The problem reads: ##f:M \rightarrow N##, and ##L \subseteq M## and ##P \subseteq N##. Then prove that ##L \subseteq f^{-1}(f(L))## and ##f(f^{-1}(P)) \subseteq P##.
My co-students and I can't find a way to prove this. I hope, someone here will be able to help us out. It would be very appreciated.

Thank you in advance and have a great day everyone.

Homework Helper
Gold Member
If this is a textbook homework type of problem, then there is a section and a format for that and we are only allowed to give hints and guidance.
Hint: pick a point in the smaller subset side and track it through the operations.

topsquark
PhysicsRock
If this is a textbook homework type of problem, then there is a section and a format for that and we are only allowed to give hints and guidance.
I guess I figured it out anyway, at least I tried. Thank you for the advice. I'll ask for a specific hint etc. next time.

Homework Helper
Gold Member
2022 Award
The problem reads: ##f:M \rightarrow N##, and ##L \subseteq M## and ##P \subseteq N##. Then prove that ##L \subseteq f^{-1}(f(L))## and ##f(f^{-1}(P)) \subseteq P##.
My co-students and I can't find a way to prove this. I hope, someone here will be able to help us out. It would be very appreciated.

Thank you in advance and have a great day everyone.
Let ##x \in L##. Then ##y = f(x) \in f(L)##. Now, what is, by definition, ##f^{-1}(f(L))##? And why is ##x \in f^{-1}(f(L))##?

Hint: it might help conceptually (be less confusing) to let ##X = f(L)## so that ##y = f(x) \in X## and show that ##x \in f^{-1}(X)##.

PS the trick with these proofs is to get all the logical steps in the right order.

FactChecker and topsquark