1. Discrete Mathematics, Part II CSE 2353 Fall 2007 Margaret H. Dunham Department of Computer Science and Engineering Southern Methodist University Some slides provided by Dr. Eric Gossett; Bethel University; St. Paul, MinnesotaSome slides provided by Dr. Eric Gossett; Bethel University; St. Paul, Minnesota Some slides are companion slides for Discrete Mathematical Structures: Theory and Applications by D.S. Malik and M.K. SenSome slides are companion slides for Discrete Mathematical Structures: Theory and Applications by D.S. Malik and M.K. Sen

2. 2 Outline  Introduction  Sets  Logic & Boolean Algebra  Proof Techniques  Counting Principles  Combinatorics  Relations,Functions  Graphs/Trees  Boolean Functions, Circuits

3. 3 Proof Technique: Learning Objectives  Learn various proof techniques Direct Indirect Contradiction Induction  Practice writing proofs  CS: Why study proof techniques?

4. 4 Proof Techniques  Theorem Statement that can be shown to be true (under certain conditions) Typically Stated in one of three ways  As Facts  As Implications  As Biimplications

5. 5 Validity of Arguments  Proof: an argument or a proof of a theorem consists of a finite sequence of statements ending in a conclusion  Argument: a finite sequence of statements.  The final statement,, is the conclusion, and the statements are the premises of the argument.  An argument is logically valid if the statement formula is a tautology.

6. 6 Proof A mathematical proof of the statement S is a sequence of logically valid statements that connect axioms, definitions, and other already validated statements into a demonstration of the correctness of S. The rules of logic and the axioms are agreed upon ahead of time. At a minimum, the axioms should be independent and consistent. The amount of detail presented should be appropriate for the intended audience.

7. 7 Proof Techniques  Direct Proof or Proof by Direct Method Proof of those theorems that can be expressed in the form ∀x (P(x) → Q(x)), D is the domain of discourse Select a particular, but arbitrarily chosen, member a of the domain D Show that the statement P(a) → Q(a) is true. (Assume that P(a) is true Show that Q(a) is true By the rule of Choose Method (Universal Generalization), ∀x (P(x) → Q(x)) is true

8. 8 Proof Techniques  Indirect Proof The implication P → Q is equivalent to the implication (  Q →  P) Therefore, in order to show that P → Q is true, one can also show that the implication (  Q →  P) is true To show that (  Q →  P) is true, assume that the negation of Q is true and prove that the negation of P is true

9. 9 Proof Techniques  Proof by Contradiction Assume that the conclusion is not true and then arrive at a contradiction Example: Prove that there are infinitely many prime numbers Proof:  Assume there are not infinitely many prime numbers, therefore they are listable, i.e. p 1,p 2,…,p n  Consider the number q = p 1 p 2 …p n +1. q is not divisible by any of the listed primes  Therefore, q is a prime. However, it was not listed.  Contradiction! Therefore, there are infinitely many primes.

10. 10 Proof Techniques  Proof of Biimplications To prove a theorem of the form ∀ x (P(x) ↔ Q(x )), where D is the domain of the discourse, consider an arbitrary but fixed element a from D. For this a, prove that the biimplication P(a) ↔ Q(a) is true The biimplication P ↔ Q is equivalent to ( P → Q) ∧ (Q → P ) Prove that the implications P → Q and Q → P are true  Assume that P is true and show that Q is true  Assume that Q is true and show that P is true

11. 11 Proof Techniques  Proof of Equivalent Statements Consider the theorem that says that statements P,Q and r are equivalent Show that P → Q, Q → R and R → P  Assume P and prove Q. Then assume Q and prove R Finally, assume R and prove P What other methods are possible?

12. 12 Other Proof Techniques  Vacuous  Trivial  Contrapositive  Counter Example  Divide into Cases  Constructive

13. 13 Proof Basics You can not prove by example

14. 14 Proof Strategies with Quantifiers  Existential Constructive  some mathematicians only accept constructive proofs Nonconstructive  show that denying existence leads to a contradiction  Universal to prove false:  one counter-example to prove true:  usually harder  the choose method

15. 15 Proofs in Computer Science  Proof of program correctness  Proofs are used to verify approaches

16. 16 Mathematical Induction  Assume that when a domino is knocked over, the next domino is knocked over by it  Show that if the first domino is knocked over, then all the dominoes will be knocked over

17. 17 Mathematical Induction  Let P(n) denote the statement that then n th domino is knocked over  Base Step: Show that P(1) is true  Inductive Hypothesis: Assume some P(i) is true, i.e. the i th domino is knocked over for some  Inductive Step: Prove that P(i+1) is true, i.e.

18. 18 Outline  Introduction  Sets  Logic & Boolean Algebra  Proof Techniques  Counting Principles  Combinatorics  Relations,Functions  Graphs/Trees  Boolean Functions, Circuits

19. 19 Learning Objectives  Learn the basic counting principles—multiplication and addition  Explore the pigeonhole principle  Learn about permutations  Learn about combinations

20. 20 Basic Counting Principles

21. 21 Basic Counting Principles

22. 22 Pigeonhole Principle The pigeonhole principle is also known as the Dirichlet drawer principle, or the shoebox principle.

23. 23 Pigeonhole Principle

24. 24 Permutations

25. 25 Permutations

26. 26 Combinations

27. 27 Combinations

28. 28 Generalized Permutations and Combinations