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## Hand Prove that G Is the Inverse Off and The Complex Functions Problems

### Question Description

I’m working on a set theory question and need support to help me learn.

Problem 1:

Let A, B be sets and let f:A?B. f is surjective if and only if for all Y? B,Y ? f(f^?1(Y)).

Problem 2:

Let G be a group, let H be a subgroup of G, and let a ? G. Let f:H?aH given by f(h) =ah^?1 for all h ? H and let g: aH?H given by g(x) =x^? a for all x ? aH .[Note that in the definition of g above, its not obvious that g is actually a function; that is, itsnot obvious that g(x)?H.]Prove that g(x)?Hf or all x?a Hand prove that g is the inverse off.

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