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



Warm-up-exercises

Let   and   denote sets. Prove the following identities.

  1.  
  2.  
  3.  
  4.  
  5.  
  6.  
  7.  
  8.  
  9.  


Prove the following (settheoretical 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.

  1.  
  2.  
  3.  


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.  ,
  5. There exists a set   such that  ,
  6. 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.