# Thread: Is f injective? Surjective? Bijective?

1. ## 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.

2. Originally Posted by Danneedshelp
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 $\displaystyle A\cap V=B\cap V$ does not necessarily imply that $\displaystyle A=B$.

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