Thread: Inverse functions of unions and intersections

Here is the problem:

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

(G H) = (G) (H)

and

(G H) = (G) (H)






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

Originally Posted by osudude

Here is the problem:

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

(G H) = (G) (H)

and

(G H) = (G) (H)



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

If then and the conclusion readily follows. So, WLOG assume that . Let then so that so . The opposite inclusion is similar.

If the conclusion is clear. So, suppose WLOG that . Let then . Clearly then we see that . Once again, the opposite inclusion is similar.

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

Originally Posted by osudude

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

I made a slight "typo". Find it for your own good.

the preserve the union, intersection, and complement of sets.