In one of the previous chapters, we have learned about basic operations on sets like for instance the set union, the $A\cup B$, or the set intersection, $A\cap B,$ or set difference $A\setminus B$. It turns out that functions always behave in a certain way in conjunction with set operations, regardless of the kind of functions and the kind of sets involved. We now want to prove some of these behavior rules.

Lemma: Behavior of Functions with Set Operations

Let $A,B$ be arbitrary sets. For arbitrary functions $f:A\mapsto B$ the following rules hold:

  1. $f[X\cup Y]=f[X]\cup f[Y]$ for all subsets of $X,Y\subseteq A.$
  2. $f[X\cap Y]\subseteq f[X]\cap f[Y]$ for all $X,Y\subseteq A.$
  3. $f^{-1}[X\cup Y]=f^{-1}[X]\cup f^{-1}[Y]$ for all $X,Y\subseteq B.$
  4. $f^{-1}[X\cap Y]=f^{-1}[X]\cap f^{-1}[Y]$ for all $X,Y\subseteq B.$
  5. $f[A\setminus X]\supseteq f[A]\setminus f[X]$ for all $X \subseteq A.$
  6. $f^{-1}[B\setminus X]=A\setminus f^{-1}[X]$ for all $X\subseteq B.$
  7. $f^{-1}[f[X]]\supseteq X$ for all $X\subseteq A.$
  8. $f^{-1}[f[X]]\supseteq X$ for all $X\subseteq A.$

Proofs: 1 Explanations: 1

Explanations: 1
Proofs: 2

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  1. Reinhardt F., Soeder H.: "dtv-Atlas zur Mathematik", Deutsche Taschenbuch Verlag, 1994, 10th Edition
  2. Wille, D; Holz, M: "Repetitorium der Linearen Algebra", Binomi Verlag, 1994