- #1
Scienticious
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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 x is in f-1(f(E)).
Since x in a implies x is in f-1(f(E)),
E is a subset of f-1(f(E)).
Conversely, let x be in f-1(f(E)).
Then f(x) is in f(E).
Since f is again injective it has an inverse; applying the inverse shows that
x is in E.
Therefore, since x in f-1(f(E)) implies that x is in E,
f-1(f(E)) is a subset of E.
Therefore, since f-1(f(E)) is a subset of E and vice versa, the two sets are equal.
QED :3
I'm pretty sure this is right, but there might be a mistake in the 2nd part. Thanks guys :3