- #1
Scienticious
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[f]^{}[/2]
Show that if f: A→B is surjective and and H is a subset of B, then f(f^(-1)(H)) = H.
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) = y.
But x in f^(-1)(H) implies that f(x) is in H, by definition of inverse functions.
Therefore f(f^(-1)(H)) is a subset of H.
Conversely, for y in H y must also be in f(f^(-1)(H)) since the latter expression is equivalent to H by definition
of inverse functions.
Since f(f^(-1)(H)) is a subset of H and vice versa, f(f^(-1)(H)) = H.
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) = y.
But x in f^(-1)(H) implies that f(x) is in H, by definition of inverse functions.
Therefore f(f^(-1)(H)) is a subset of H.
Conversely, for y in H y must also be in f(f^(-1)(H)) since the latter expression is equivalent to H by definition
of inverse functions.
Since f(f^(-1)(H)) is a subset of H and vice versa, f(f^(-1)(H)) = H.
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