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Math Help - Is f injective? Surjective? Bijective?

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    Senior Member Danneedshelp's Avatar
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    Is f injective? Surjective? Bijective?

    Let U be an universe, and V⊆U . Define a function f:P(U)→P(V), by f(A)=A∩V, for all A∈P(U). Is f injective? Surjective? Bijective? Fully justify your answers.

    Not really sure where to start with this one. I would think its injective since, if I let f(A)=f(B) then A∩V=B∩V. This would mean B=A. But, I think I would have to do cases, because B can't always be equal to A....? I'm not sure what I'm doing.

    Some help would be appreciated.
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    Quote Originally Posted by Danneedshelp View Post
    Let U be an universe, and V⊆U . Define a function f:P(U)→P(V), by f(A)=A∩V, for all A∈P(U). Is f injective? Surjective? Bijective? I would think its injective since, if I let f(A)=f(B) then A∩V=B∩V. This would mean B=A. But, I think I would have to do cases, because B can't always be equal to A....? .
    If A\cap V=B\cap V does not necessarily imply that A=B.

    Because V \subseteq U then W \in P(V)\; \Rightarrow \;W \subseteq V\; \Rightarrow \;f(W) = W \cap V = W.
    Does that show that f is surjective?
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