O I understand now I think, thank you :3
f is surjective, so for any y in H we have f(x) = y.
since f^(-1)(H) is in the domain of f by definition of inverse functions,
y = f(f^(-1)(y)) for any y in H and thus H is a subset of f(f^(-1)(H)).
This line of reasoning checks out, right?
[f]^{}[/2]Homework Statement
Show that if f: A→B is surjective and and H is a subset of B, then f(f^(-1)(H)) = H.
Homework Equations
The Attempt at a Solution
Let y be an element of f(f^(-1)(H)).
Since f is surjective, there exists an element x in f^(-1)(H) such that f(x) =...
Homework Statement
Suppose that f is an injection. Show that f-1(f(x)) = x for all x in D(f).
Homework Equations
The Attempt at a Solution
Let z be in f-1(f(x)).
Then f(z) is in f(x) by definition of inverse functions.
Since f is injective, z = x for some x in D(f).
Thus...
O of course, since f(x) is in of f(E), then x must be an element of E because injectivity implies that each value of f(x1) corresponds to a unique element of A.
I think this works, but at any rate I'm still stuck on the other half of the proof..
if I try to suppose x is in E I get back to the...
O dear I see my mistake; I confused the definitions of injective and bijective :/
Then for the part of the proof you mentioned,
x is in f-1(f(E)).
Thus f(x) is in f(E).
Then it seems obvious that x is in E at this point, but I'm not sure if it's trivial or not.
So if this step works we...
Homework Statement
Show that if f: A → B is injective and E is a subset of A, then f −1(f(E) = E
Homework Equations
The Attempt at a Solution
Let x be in E.
This implies that f(x) is in f(E).
Since f is injective, it has an inverse.
Applying the inverse function we see that...
Thank you for your reply :3 I think I get it now; here's my new proof:
Let x be an element of E.
Then f(x) is an element of f(E).
But if x is an element of E then x is an element of E ∪ F
Thus f(x) is an element of f(E ∪ F).
Therefore f(E) is a subset of f(E ∪ F).
Now let x be an element of...
Homework Statement
Show that if f: A → B and E, F are subsets of A, then f(E ∪ F) = f(E) ∪ f(F).
Homework Equations
The Attempt at a Solution
My attempt:
Suppose x is an element of E. Then f(x) is an element of f(E), which means f(x) is a subset of f(E).
But x is in E...