G of f injective, but g not injective

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Homework Help Overview

The discussion revolves around finding two functions, \( f: A \rightarrow B \) and \( g: B \rightarrow C \), such that the composition \( g \circ f \) is injective while \( g \) itself is not injective. This involves concepts from function composition and properties of injective functions.

Discussion Character

  • Exploratory, Conceptual clarification

Approaches and Questions Raised

  • Participants express uncertainty about the requirements of the problem and whether it involves a trial-and-error approach. There is a suggestion to consider the injectivity of \( g \) specifically on the image of \( f \).

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem. Some guidance has been offered regarding the relationship between the injectivity of \( g \) and the image of \( f \>.

Contextual Notes

Participants are navigating the definitions of injective functions and the implications of function composition. There may be assumptions about the nature of the functions involved that are not fully articulated.

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Homework Statement


Give an example of a map f:A[tex]\rightarrow[/tex]B and a map g:B[tex]\rightarrow[/tex]C where g of f is injective but g is not injective.


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The Attempt at a Solution


I'm not really sure what they are asking for.
 
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In a nutshell, find two functions, f and g, so that g [itex]\circ[/itex] f is one-to-one, but f [itex]\circ[/itex] g is not.
 
So it's basically just guess and check?
 
It doesn't have to be. Think about arranging it so that g is injective on the image of f but not on the entire domain of g.
 

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