Is f-1(A ∪ B) the same as f-1(A) ∪ f-1(B)?

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Homework Help Overview

The discussion revolves around the properties of inverse images in set theory, specifically questioning whether the inverse image of the union of two sets is equal to the union of the inverse images of the individual sets.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the validity of the statement by suggesting proofs and discussing the method of proving set equality through subset relationships.

Discussion Status

Some participants have offered guidance on how to approach proving the statement, while others express uncertainty about the mathematical representation involved. Multiple lines of reasoning are being explored without a clear consensus on the proof's completeness.

Contextual Notes

There is mention of constraints related to the scope of the participants' class, which may limit the depth of mathematical rigor they can apply in their proofs.

brookey86
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Yes, you are. But I'd feel more comfortable if you tried to prove it rather than taking as an assumption.
 
Have you tried to prove it's true?
 
SammyS said:
Have you tried to prove it's true?

I can prove it using words, not quite there using mathematical symbols, but that part is out of the scope of my class. Thanks guys!
 
You prove two sets are equal by proving that each is a subset of the other. You prove "A" is a subset of "B" by saying "let x\in A", then show "x\in B".

Here, to show that f^{-1}(A\cup B)\subset f^{-1}(A)\cup f^{-1}(B), start by saying "let x\in f^{-1}(A\cup B)". Then y= f(x)\in A\cup B. And that, in turn, means that either y\in A or y \in B. Consider each of those.

Note, by the way, that we are considering the inverse image of sets. None of this implies or requires that f actually have an "inverse".
 

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