# Thread: Inverse functions of unions and intersections

1. ## Inverse functions of unions and intersections

Here is the problem:

show that if f:A $\rightarrow$ B and G, H are substes of B, then

$f^{-1}$(G $\cup$H) = $f^{-1}$ (G) $\cup$ $f^{-1}$(H)

and

$f^{-1}$(G $\cap$H) = $f^{-1}$ (G) $\cap$ $f^{-1}$(H)

So I understand the main principle behind an inverse, such that if f(x) $\in$H, then x $\in f^{-1}$(H). I also understand the proof for inverse composite functions. But this is confusing. How do I prove this?? Thank you

2. Originally Posted by osudude
Here is the problem:

show that if f:A $\rightarrow$ B and G, H are substes of B, then

$f^{-1}$(G $\cup$H) = $f^{-1}$ (G) $\cup$ $f^{-1}$(H)

and

$f^{-1}$(G $\cap$H) = $f^{-1}$ (G) $\cap$ $f^{-1}$(H)

So I understand the main principle behind an inverse, such that if f(x) $\in$H, then x $\in f^{-1}$(H). I also understand the proof for inverse composite functions. But this is confusing. How do I prove this?? Thank you
If $G\cup H=\varnothing$ then $G=H=\varnothing$ and the conclusion readily follows. So, WLOG assume that $G,H\ne\varnothing$. Let $x\in f^{-1}\left(G\cup H\right)$ then $f(x)\in \left(G\cup H\right)$ so that $f(x)\in G\text{ or }f(x)\in H$ so $x\in f^{-1}\left(G\right)\text{ or }f^{-1}(x)\in H\Longleftrightarrow x\in\left( f^{-1}\left(G\right)\cup f^{-1}\left(H\right)\right)$. The opposite inclusion is similar.

If $G\cap H=\varnothing$ the conclusion is clear. So, suppose WLOG that $G\cap H\ne\varnothing$. Let $x\in f^{-1}\left(G\cap H\right)$ then $f(x)\in \left(G\cap H\right)\Longleftrightarrow f(x)\in G\text{ and }f(x)\in H$. Clearly then we see that $x\in f^{-1}\left(G\right)\text{ and }x\in f^{-1}\left(H\right)\Longleftrightarrow x\in\left(f^{-1}\left(G\right)\cap f^{-1}\left(H\right)\right)$. Once again, the opposite inclusion is similar.

3. haha seems so easy now!!! thank you soo much!

4. Originally Posted by osudude
haha seems so easy now!!! thank you soo much!

5. the $f^{-1}$ preserve the union, intersection, and complement of sets.

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### prove the inverse of a union b is the same as the inverse of a union the inverse of b

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