- #1

CGandC

- 326

- 34

- Homework Statement
- We define the function ## G = \lambda A \in P(Y). f^{-1}[A] ## ,

Prove that for every ## X , Y ## and ## f \in X \rightarrow Y ## :

## G ## is Injective ##\iff ## ## f ## is onto ## Y ##.

- Relevant Equations
- -Familiarity with Lambda notation ( from lambda calculus )

- ## P(Y) ## is power-set of ## Y ##

**My attempt:**

## ( \rightarrow ) ## Suppose G is injective. Let ## y \in Y ## be arbitrary, denote A = ## \{ y \} ## so that ## G(A) = G(\{ y \}) = f^{-1}[\{ y \}] = \{ x \in X | f(x) \in \{ y \} \} =\{ x \in X | f(x)= y \} ##.

[ However, now I am stuck because I don't know if ## G(A)= \emptyset## and trying to assume it and find a contradiction also had me getting stuck. If I knew that ## G(A) \neq \emptyset ## I'm practically finished proving this side. Any ideas? ].

## ( \leftarrow ) ## Suppose ## f ## is onto ## Y ##. Let ## A_1 , A_2 \in P(Y) ## be arbitrary. Suppose ## G(A_1) = G(A_2) ## , meaning ## \{ x \in X | f(x) \in A_1 \} = \{ x \in X | f(x) \in A_2 \} ##

[ I'm kinda lost, how do I use the assumption that f is onto in order to continue? ]

Thanks in Advance!