Proving Sets and Functions Homework: f(f^-1(C)) = [C Intersection Im(f)]

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Homework Help Overview

The discussion revolves around proving a set equality involving a function and its inverse. The original poster presents a function f from set A to set B and a subset C of B, seeking to prove that f(f^-1(C)) equals the intersection of C and the image of f.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the method of proving set equality by showing mutual inclusion. There is a focus on demonstrating that an element y in f(f^-1(C)) also belongs to C intersection Im(f), and vice versa. Questions arise about the specific properties that y must satisfy to establish this inclusion.

Discussion Status

The conversation is ongoing, with participants providing guidance on how to approach the proof without using visual aids like Venn diagrams. Some participants suggest writing down properties of y to aid in the proof, indicating a productive direction in the discussion.

Contextual Notes

There is a request for general tips on approaching problems related to sets and functions, indicating a potential lack of familiarity with the topic. The original poster expresses a desire for clarification on specific elements involved in the proof.

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Homework Statement



f: A -> B is a function with C a subset of B. Prove that
f(f^-1(C)) = [C intersection Im(f)]. (f^-1(c) = f inverse of C)

Homework Equations





The Attempt at a Solution



Please let me know how to approach to the solution (not using venn diagrams). Also if possible give me any link that gives tips on approaching any general sets and functions problem.
Thank you.
 
Last edited:
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What's K?
 
By K did you mean C?

Usually when you want to prove two sets are equal to each other, you prove each is a subset for the other. If [tex]y\in f(f^{-1}(C))[/tex] can you prove that [tex]y\in C\cap Im(f)[/tex]? Then you would have that [tex]f(f^{-1}(C))\subset C\cap Im(f)[/tex]. Do the other way also and you're done
 
I am sorry, that C.. how can I prove that y belongs to c intersection im(f)?
 
Well, you have some property that y satisfies because we know it belongs to [tex]f(f^{-1}(C))[/tex], which you should try to write down. Then use it
 

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