Surjection: Is f^-1(X) Surjective? Why?

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Discussion Overview

The discussion centers around the concept of surjectivity in functions, specifically whether the inverse image of a set under a surjective function is also surjective. Participants explore the definitions and implications of surjective functions and their inverse images, with a focus on the function f: N-->X.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions if f^-1(X) is surjective given that f is surjective.
  • Another participant points out that f^-1(X) is not a function unless f is bijective, suggesting a misunderstanding of the terms involved.
  • A different participant clarifies that f^-1(X) represents a set of natural numbers and is not a function, which leads to further interpretation of the question posed.
  • One participant proposes that if f: N->X is surjective, then f^-1(X) equals N, arguing that surjectivity does not affect this equality.

Areas of Agreement / Disagreement

Participants express differing interpretations of the original question and the nature of f^-1(X). There is no consensus on whether f^-1(X) can be considered surjective, as the discussion reveals multiple competing views on the definitions and implications of surjectivity and inverse images.

Contextual Notes

There are limitations in the discussion regarding the assumptions about the nature of functions and their inverse images. The dependence on definitions of surjectivity and the interpretation of f^-1(X) as a set rather than a function remains unresolved.

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If a function f: N-->X is surjective , is f^-1(X) (its inverse image) also surjective? If so, why?
 
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f^-1(X) isn't a function...

Anyways, have you looked at any examples?
 
Hurkyl meant: isn't a function unless f is one-one, i.e. injective and surjective, i.e. bijective. Was this a trick question from some problem set?
 
No, Chris, I don't believe that's what Hurkyl meant! I started to interpret f-1(X) as if it were f-1(x) and say "that's not necessarily a function", but f-1(X) is the "inverse image" of X. It's not a function for the very good reason that f-1(X) is a set of natural numbers.

I suspect that the correct question was "If f: N->X is surjective is f-1(X)= N?" If I understand what is meant by "f:N->X", then "surjective" is irrelevant. For ANY function f:N->X, that is, "to every point in N assigns a point in X", f-1(X)= N.
 

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