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Show that the set of all finite subsets ofX is countable

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EXERCISES 8: LECTURE FOUNDATIONS OF MATHEMATICS

Exercise 1. Let X and Y be finite sets. Decide (with a proof) how many injective maps X→Y exist.

Exercise 2. Show that a setX6=∅ is countable if and only if there is a surjection fromN0 to X.

Exercise 3. LetX be a countable set. Show that the set of all finite subsets ofX is countable.

Exercise 4. LetX be a set. Show that the following statements are equivalent.

(i) X is infinite.

(ii) For all mapsf:X→X there exists ∅(A(X withf(A)⊂A.

Hint: Takef:{0,1, . . . , n} → {0,1, . . . , n}, f(i) =i+ 1wheren+ 1should be considered as0.

Does it satisfy (ii)? Moreover, show that (ii) holds forX=N0 and reduce the general case to this situation.

Submission of the exercise sheet: 11.Nov.2019 before the lecture. Return of the exercise sheet: 14.Nov.2019 during the exercise sessions.

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