Why is the following not true: f(f^(-1)(B))=B?

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SUMMARY

The statement $f(f^{-1}(B))=B$ is not universally true due to the potential lack of surjectivity in the function $f:A\to B$. If the function $f$ does not cover the entire set $B$, then the composition $f(f^{-1}(B))$ results in $f(A)$, which is a subset of $B$ and not necessarily equal to $B$. This highlights the importance of understanding the properties of functions, particularly surjectivity, in mathematical analysis.

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  • Understanding of function properties, specifically surjectivity.
  • Familiarity with inverse functions and their definitions.
  • Basic knowledge of set theory and function notation.
  • Concept of range and domain in mathematical functions.
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OhMyMarkov
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The title says it: why is the following not true: $f(f^{-1}(B))=B$?

Thanks!
 
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Re: Why is the following not true: $f(f^{-1}(B))=B$?

OhMyMarkov said:
The title says it: why is the following not true: $f(f^{-1}(B))=B$?

Thanks!
You have not given a context for this, but in general if you have a function $f:A\to B$ there is no reason to suppose that the range of $f$ is the whole of $B$. If $f$ is not surjective then $f(f^{-1}(B)) = f(A) \ne B$.
 

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