1. BAI CM20144 Applications I: Mathematics for Applications Mark Wood cspmaw@cs.bath.ac.uk http://www.cs.bath.ac.uk/~cspmaw

2. BAI Proofs Preparation Methods Marks General Coursework Questions Matrix Proofs: Examples Test 4 Todays Tutorial

3. BAI Proofs: Preparation

4. BAI Break down the proof into bits Make sure youve got everything covered (ie. iff) Proofs: Preparation

5. BAI Break down the proof into bits Make sure youve got everything covered (ie. iff) Write down everything you know What you can assume (ie. where youre starting from) What youre aiming for Any definitions that could be useful Any results you could use Proofs: Preparation

6. BAI Break down the proof into bits Make sure youve got everything covered (ie. iff) Write down everything you know What you can assume (ie. where youre starting from) What youre aiming for Any definitions that could be useful Any results you could use Remember: you know everything you need Just need to use it Proofs: Preparation

7. BAI Proofs: Methods 1

8. BAI Proof by implication A I 1 I 2 … B Direction of implication is fixed BUT can work from both ends Proofs: Methods 1

9. BAI Proof by implication A I 1 I 2 … B Direction of implication is fixed BUT can work from both ends Proof by contradiction A and not(B) C 1 C 2 … not(A) Often simpler than proof by implication Often easier to justify each step Proofs: Methods 1

10. BAI Proofs: Methods 2

11. BAI Proof by induction Works when youve got to prove sth for all n Prove its true for an easy case Assume its true for case n Prove its then also true for case n+1 Domino effect Proofs: Methods 2

12. BAI Proof by induction Works when youve got to prove sth for all n Prove its true for an easy case Assume its true for case n Prove its then also true for case n+1 Domino effect Proof by example Write down an example for which it works Quick and simple Proofs: Methods 2

13. BAI Proof by induction Works when youve got to prove sth for all n Prove its true for an easy case Assume its true for case n Prove its then also true for case n+1 Domino effect Proof by example Write down an example for which it works Quick and simple NOT A PROOF Proofs: Methods 2

14. BAI Proof by induction Works when youve got to prove sth for all n Prove its true for an easy case Assume its true for case n Prove its then also true for case n+1 Domino effect Proof by example Write down an example for which it works Quick and simple Proofs: Methods 2

15. BAI Proof by induction Works when youve got to prove sth for all n Prove its true for an easy case Assume its true for case n Prove its then also true for case n+1 Domino effect Proof by counterexample Works when trying to show sth does not hold Construct an example where it does hold Special case of proof by contradiction Proofs: Methods 2

16. BAI Proofs: Marks

17. BAI Shortness As little redundancy of working as possible As little pointless waffle as possible Proofs: Marks

18. BAI Shortness As little redundancy of working as possible As little pointless waffle as possible Clarity State everything you do clearly and concisely Lay out your work well! Proofs: Marks

19. BAI Shortness As little redundancy of working as possible As little pointless waffle as possible Clarity State everything you do clearly and concisely Lay out your work well! Generality Dont make unwarranted assumptions Dont try and use proof by example Proofs: Marks

20. BAI Proofs: Marks Summary

21. BAI Marked by humans Make it as easy as possible to mark Were not total imbeciles Were not psychic Proofs: Marks Summary

22. BAI Marked by humans Make it as easy as possible to mark Were not total imbeciles Were not psychic For full marks… Prove totally general results using as few steps as possible. Concisely explain each step you take in your proof. State every definition and theorem you use. Use neat and ordered layout. Proofs: Marks Summary

23. BAI General Coursework Questions

24. BAI Ex. Set 2.1, q. 15 – 17 Ex. Set 2.2, q. 8, 14 – 17, 21 – 29, 33 – 34 Ex. Set 2.3, q. 4 – 14, 16 – 20 Ex. set 2.4, q. 12, 20 – 26 Chapter 2 review ex., q. 10 – 18 Ex. set 3.2, q. 11 – 21 Ex. set 3.3, q. 8 – 15 Ex. set 3.4, q. 18 – 27 Chapter 3 review ex., q. 9 – 13, 16 – 18 Matrix Proofs: Examples