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Theorem (Image And Preimage Properties)

Consider the function $f:X\to Y$.

  1. If $A\subseteq X$, then we have $A\subseteq f^{-1}(f(A))$.
  2. If $B\subseteq Y$, then we have $f(f^{-1}(B))\subseteq B$.
  3. If $A_1\subseteq X$ and $A_2\subseteq X$, then we have $f(A_1\cap A_2)\subseteq f(A_1)\cap f(A_2)$.
  4. If $A_1\subseteq X$ and $A_2\subseteq X$, then we have $f(A_1\cup A_2)=f(A_1)\cup f(A_2)$.
  5. If $B_1\subseteq Y$ and $B_2\subseteq Y$, then we have $f^{-1}(B_1\cap B_2)=f^{-1}(B_1)\cap f^{-1}(B_2)$.
  6. If $B_1\subseteq Y$ and $B_2\subseteq Y$, then we have $f^{-1}(B_1\cup B_2)=f^{-1}(B_1)\cup f^{-1}(B_2)$.
  7. We have $f(A)\subseteq B$ if and only if $A\subseteq f^{-1}(B)$.
  8. If $A_1\subseteq A_2\subseteq X$, then we have $f(A_1)\subseteq f(A_2)$.
  9. If $B_1\subseteq B_2\subseteq Y$, then we have $f^{-1}(B_1)\subseteq f^{-1}(B_2)$.
  10. If $B\subseteq Y$, then $f^{-1}(Y\setminus B)=X\setminus f^{-1}(B)$.