Explanation: How Functions interact with Set Operations

(related to Lemma: Behavior of Functions with Set Operations)

In the above lemma, the rules $2, 5,7$ and $8$ do not allow equality in the general case. The proof given for the lemma is simple but highly technical and it might be interesting to see a counterexample. Such a counterexample is given below.


In the figure, a function $f:A\mapsto A$ is defined and where we set $X:=\{u\}$ and $Y:=\{w\}.$ We calculate the above rules to see that, indeed, no equality can be achieved:

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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