Assume that F: A to B is a function and S, a subset of B, is a subset. Prove that f(f-1(S))= S intersected with f(A).

[ f(f-1)) is f and inverse of f]

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- Nov 28th 2005, 01:12 PMummmmhelp, i'd be impressed
Assume that F: A to B is a function and S, a subset of B, is a subset. Prove that f(f-1(S))= S intersected with f(A).

[ f(f-1)) is f and inverse of f] - Nov 28th 2005, 01:42 PMCaptainBlackQuote:

Originally Posted by**ummmm**

Or have I missed something.

Of course I have missed something:

choose to be ,

then:

-the empty set.

Back to the problem:

(we need only consider the case where as the result holds trivally otherwise)

,

so if then and

and ,

i.e.

Now suppose then clearly

and so

hence:

RonL - Nov 28th 2005, 02:48 PMummmmi think f(a)=b
i think f(a) has to equal b because there if a function from a to b.

- Nov 28th 2005, 05:00 PMummmmanyone?
anyone?

- Nov 29th 2005, 12:40 AMlewisje
B is merely the codomain for f

f(A) may be a proper subset of B

All that is necessary is that if x and y are in A and f(x) and f(y) are distinct, then x and y are distinct. - Dec 1st 2005, 09:07 PMCaptainBlack
**The proof on its own:**

(we need only consider the case where as the result holds trivally otherwise)

,

so if then and

and ,

i.e.

Now suppose then clearly

and so

hence:

RonL