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Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 1

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Warm-up-exercises

Let and denote sets. Prove the following identities.


Prove the following (set-theoretical versions of) syllogisms of Aristotle. Let denote sets.

  1. Modus Barbara: and imply .
  2. Modus Celarent: and imply .
  3. Modus Darii: and imply .
  4. Modus Ferio: and imply .
  5. Modus Baroco: and imply .


Prove the following formulas by induction.


Show that (with being the only exception) the relation

holds.


Show, by induction, that for every , the number

is a multiple of .


Prove, by induction, that the following inequality holds


Prove, by induction, that the formula

holds for all .


The cities are connected by roads, and there is exactly one road between each couple of cities. Due to construction works, at the moment all roads are drivable only in one direction. Show that nevertheless, there exists one city from which you can reach all the others.




Hand-in-exercises

Exercise (4 marks)

Let and be sets. Show that the following facts are equivalent.

  1. ,
  2. ,
  3. ,
  4. There exists a set such that ,
  5. There exists a set such that .


Exercise * (3 marks)

Prove, by induction, that the sum of consecutive odd numbers (starting from ) is always a square number.


Exercise (3 marks)

Fix . Show, by induction, that the following identity holds.


Exercise (4 marks)

An -chocolate is a rectangular grid, which is divided by longitudinal grooves and by transverse grooves into smaller bite-sized rectangles. A dividing step of a chocolate is the complete severing of a chocolate, along a longitudinal or a transverse groove. A complete breakdown of a chocolate is a consequence of division steps (each one applied to a previously obtained intermediate chocolate), whose final product consists of all the small bite-sized pieces, more handy to be eaten. Show, by induction, that each breakdown of an -chocolate consists of exactly division steps.