Proof about pre-images of functions

In summary, the problem involves proving that the subsets ##L## and ##P## are contained within the preimage and image of a given function ##f:M \rightarrow N##. After struggling to find a solution, the students seek help and are advised to pick a point and track it through the operations. Eventually, the students are able to figure out the solution and are reminded to ask for specific hints in the future. The solution involves letting ##X = f(L)## and showing that ##x \in f^{-1}(X)## through logical steps.
  • #1
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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.
 
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  • #2
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.
 
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  • #3
FactChecker said:
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.
 
  • #4
PhysicsRock said:
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.
 
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1. What is a pre-image of a function?

A pre-image of a function is the set of all input values that produce a specific output value when plugged into the function. It can also be thought of as the original set of values before they are transformed by the function.

2. How do you prove that a set of values is a pre-image of a function?

To prove that a set of values is a pre-image of a function, you must show that when those values are input into the function, they produce the desired output. This can be done by plugging in each value and showing that the resulting output matches the given output.

3. Can a pre-image of a function have multiple images?

Yes, a pre-image of a function can have multiple images. This means that there can be multiple input values that produce the same output value when plugged into the function.

4. What is the relationship between a pre-image and an inverse function?

A pre-image and an inverse function are closely related. The pre-image is the original set of values before being transformed by the function, while the inverse function is the function that undoes the original transformation. In other words, the pre-image of a function is the range of the inverse function.

5. How is proving pre-images useful in mathematics?

Proving pre-images is useful in mathematics because it allows us to verify the validity of a function. It also helps us understand the relationship between the input and output values of a function, and can be used to find the inverse function. Additionally, proving pre-images is essential in many areas of mathematics, such as calculus and linear algebra.

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