# Is the solution to this problem as trivial as I think?

• I
• PhysicsRock
In summary, the problem involves sets and a function ##f## with subsets ##L## and ##P##. The task is to prove that ##L## is a subset of the preimage of the image of ##L## under ##f##, and that the preimage of the image of ##P## under ##f## is a subset of ##P##. This can be easily shown using the definition of a function's preimage, as shown in the conversation. However, there is a typo in the statement involving ##P##, and ##f^{-1}## is not the inverse function but a set of preimages.
PhysicsRock
The problem goes as follows: Let ##M, N## be sets and ##f : M \rightarrow N##. Further let ##L \subseteq M## and ##P \subseteq N##. Then show that ##L \subseteq f^{-1}(f(L))## and ##f^{-1}(f(P)) \subseteq P##.
Obviously, I would simply use the definition of a functions inverse to obtain ##f^{-1}(f(L)) = L \subseteq L## and vice versa for ##P##. This seems quite trivial to me though, so am I doing this correctly or is there a mistake in my thoughts?

Thank you everyone and have a great day.

PhysicsRock said:
The problem goes as follows: Let ##M, N## be sets and ##f : M \rightarrow N##. Further let ##L \subseteq M## and ##P \subseteq N##. Then show that ##L \subseteq f^{-1}(f(L))## and ##f^{-1}(f(P)) \subseteq P##.
Obviously, I would simply use the definition of a functions inverse to obtain ##f^{-1}(f(L)) = L \subseteq L## and vice versa for ##P##. This seems quite trivial to me though, so am I doing this correctly or }1is there a mistake in my thoughts?

Thank you everyone and have a great day.
You have a typo in the ##P## statement.

##f^{-1}## is not the inverse function. It is a set, namely ##f^{-1}(L)=\{m\in M\,|\,f(m)\in L\}.##

It isn't difficult to prove these statements, but it's not about inverse functions, just preimages.

fresh_42 said:
You have a typo in the ##P## statement.

##f^{-1}## is not the inverse function. It is a set, namely ##f^{-1}(L)=\{m\in M\,|\,f(m)\in L\}.##

It isn't difficult to prove these statements, but it's not about inverse functions, just preimages.
That clears things up, thank you :)

As an example if you're still confused, ##f:\mathbb{,R} \to \mathbb{R}## defined by ##f(x)=x^2##. Try computing ##f^{-1}(f(L))## for ##L=[0,1]##

## 1. Is there a simple solution to this problem?

The answer to this question depends on the specific problem at hand. Some problems may have straightforward solutions, while others may require more complex approaches. It is important to carefully analyze the problem and determine the most appropriate solution.

## 2. Can the problem be solved quickly and easily?

Again, this depends on the nature of the problem. Some problems may have quick and easy solutions, while others may require more time and effort to solve. It is important to thoroughly understand the problem and consider all possible solutions before determining the best course of action.

## 3. Are there any potential complications that I should be aware of?

It is always important to consider potential complications when solving a problem. Even seemingly simple problems can have hidden complexities that may impact the solution. It is important to carefully analyze the problem and consider all potential obstacles before determining the best approach.

## 4. Is there any existing research or solutions that can help with this problem?

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## 5. How can I determine if my solution is the most optimal?

There are various methods for determining the optimality of a solution, such as conducting tests, analyzing data, and seeking feedback from other experts in the field. It is important to continuously evaluate and improve upon the solution to ensure it is the most effective and efficient approach.

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