- #1
roam
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Homework Statement
Let A and B be sets and let [tex]f: A \rightarrow B[/tex] be a function. Define a function
[tex]h: \mathcal{P} (B) \rightarrow \mathcal{P} (A)[/tex] by declaring that for [tex]Y \in\mathcal{P} (B)[/tex], [tex]h(Y)= \{ x \in A: f(x) \notin Y \}[/tex]. Show that if [tex]f[/tex] is a bijection then h is a bijection.
The Attempt at a Solution
I'm not quite sure where to start. I could start by first showing that f is one to one & onto and then show that f = h by showing that:
dom(f) = dom(h)
cod(f) = cod(h)
And, [tex]\forall x \in dom(f), f(x)=h(x)[/tex]
Of course, I can't do this because the question doesn't define function f (it only gived domain & codomain)! Does anyone know how to prove this question?
P.S. the notation [tex]\mathcal{P} (A)[/tex] and [tex]\mathcal{P} (B)[/tex] are supposed to represent the power set of A & B.
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