1. 1 Mathematical background – Part 1 Miloš Nováček Chair of Programming Methodology milos.novacek@inf.ethz.ch

2. 2 Classes  Every Tuesday 1pm – 2pm (starting 1.15pm)  Hand in your solution to me via email milos.novacek@inf.ethz.ch by Monday 9am milos.novacek@inf.ethz.ch  Homework is optional but helpful  You can submit also partial solutions  Master solutions and slides from classes will be published after each class on the course web-page http://www.pm.inf.ethz.ch/education/courses/spa/ex ercises http://www.pm.inf.ethz.ch/education/courses/spa/ex ercises  Feedback!!!

3. 3 Any questions about the content of the last lecture?

4. 4 Exercise 1

5. 5

6. 6 Exercise 1 - proof

7. 7 Exercise 1 – proof (cont’d)

8. 8 Exercise 2

9. 9 Inverse relation

10. 10 Exercise 2 - proof

11. 11 Exercise 2 – proof (cont’d)

12. 12 Exercise 3

13. 13 Exercise 3 - proof Can also be proven by contradiction.

14. 14 Questions?