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Problem 58: (Image And Preimage Properties 1 And 2)

Prove properties 1 and 2 for images and preimages. So prove for a function $f:X\to Y$ both of the following.

  1. If $A\subseteq X$, then $A\subseteq f^{-1}(f(A))$.
  2. If $B\subseteq Y$, then $f(f^{-1}(B)\subseteq B$.

Then give an example of a function $f:X\to Y$ and subsets $A\subseteq X$ and $B\subseteq Y$ where $A\neq f^{-1}(f(A))$ and $B\neq f(f^{-1}(B))$.

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