1. Lecture 1 Overview Topics 1. Proof techniques: induction, contradiction Proof techniques June 1, 2015 CSCE 355 Foundations of Computation

2. – 2 – CSCE 355 Summer 2015 Models of Computation

3. – 3 – CSCE 355 Summer 2015 Summer Schedule  Tests on Mondays June 8 June 15 June 22 1 hour long  Exam June 26

4. – 4 – CSCE 355 Summer 2015 Course Outcomes  Mathematical prerequisites: functions, relations, properties of relations, posets.  Proof Techniques  Finite automata: regular languages, regular expressions, DFAs, NFAs, equivalences.  Limitations: pumping lemma  Context free languages: grammars, push-down automata  Turing machines: undecidability, the halting problem  Intractability: NP, NP-Completeness

5. – 5 – CSCE 355 Summer 2015 Prerequisites  CSCE 211 Number systems, Boolean algebra, logic design, sequential machines Mealy machines Moore machines

6. – 6 – CSCE 355 Summer 2015 Prerequisites  CSCE 350 Techniques for representing and processing information, including the use of lists, trees, and graphs; analysis of algorithms; sorting, searching, and hashing techniques.  MATH 374 Propositional and predicate logic; proof techniques; recursion and recurrence relations; sets, combinatorics, and probability; functions, relations, and matrices; algebraic structures.

7. – 7 – CSCE 355 Summer 2015 Review of Relations on Sets  Binary relations - (X, Y) ἐ R or X Rel Y < on integers likes (X,Y)  Unary relation - properties boring(matthews)  Ternary relation “X was introduced to Y by Z” -- ( X, Y, Z) Table in a relational database

8. – 8 – CSCE 355 Summer 2015 Special types of Relations  Injections  Surjections  Functions

9. – 9 – CSCE 355 Summer 2015 Properties of Relations PropertyDefExampleNeg-Example Reflexive Irreflexive symmetric antisymmetric asymmetric transitive Total Injection Surjection function

10. – 10 – CSCE 355 Summer 2015 Posets  Partially Ordered Sets (POSETS) Reflexive Antisymmetric Transitive  Hasse Diagram  Topological sorting

11. – 11 – CSCE 355 Summer 2015 Equivalence relations

12. – 12 – CSCE 355 Summer 2015 Proof Techniques 1.1 Direct proof 1.2 Proof by induction 1.3 Proof by transposition 1.4 Proof by contradiction 1.5 Proof by construction 1.6 Proof by exhaustion 1.7 Probabilistic proof 1.8 Combinatorial proof 1.9 Nonconstructive proof 1.10 Proof nor disproof 1.11 Elementary proof

13. – 13 – CSCE 355 Summer 2015 Deductive Proofs the conclusion is established by logically combining the axioms, definitions, and earlier theorems Example: The sum of two even integers is even. Hypothesis

14. – 14 – CSCE 355 Summer 2015 Theorem 1.3 used to prove Theorem 1.4  Theorem 1.3 If x >= 4 then 2 x >= x 2.  Theorem 1.4 If x is the sum of the squares of 4 positive integers then 2 x >= x 2.  Proof

15. – 15 – CSCE 355 Summer 2015 Theorem 1.3 If x >= 4 then 2 x >= x 2.  f(x) = x 2 / 2 x.  Then what is the derivative f’ of f  Derivative of quotient?? http://www.math.hmc.edu/calculus/tutorials/quotient_rule/  So f’(x) =

16. – 16 – CSCE 355 Summer 2015 Proofs about Equality of Sets  To prove S = T Show S is a subset of T, and T is a subset of S  Commutative law of union  Theorem 1.10 Distributive law of union over intersection  Proof

17. – 17 – CSCE 355 Summer 2015 Proof by Contradiction

18. – 18 – CSCE 355 Summer 2015 If and only If statements  IF H then C H = Hypothesis C = conclusion H implies C H only if C C if H  A if and only if B If part : Only-if part  Theorm 1.7 ceiling = floor  x is an integer

19. – 19 – CSCE 355 Summer 2015 Induction  Given a statement S(n) about an integer n that we want to prove.  Basis Step: Show S(i) is true for a particular integer i Usually i = 0 or i = 1  Inductive Step: Assume S(n) is true for n >= i and then show S(n+1) is true  Inductive Hypothesis: Assume S(n) is true

20. – 20 – CSCE 355 Summer 2015 Example Induction Proof: Theorem 1.16

21. – 21 – CSCE 355 Summer 2015 Number of leaves in complete tree of height h is 2 h.

22. – 22 – CSCE 355 Summer 2015 More general induction  Basis step as before  Assume S(k) for all k <= n then show S(n)

23. – 23 – CSCE 355 Summer 2015 Recursive Def of Tree  Basis: a single node is a tree.  If T1, T2, … Tk are trees then a new tree can be formed by 1. Add new node N, the root of the new tree 2. Add copies of T1… Tk 3. Add an edge from N to the root of each T1, T2, … Tk

24. – 24 – CSCE 355 Summer 2015 Structural Induction  For objects with recursive definitions consisting of base objects and then combining rules  Basis step: show the proposition S(X) holds for every base object X.  Inductive step: Given a recursive structure X formed from X1, X2, … Xn by the application of the def. then Assume S(X1) S(X2) …. S(Xn) are true and show that S(X) is true

25. – 25 – CSCE 355 Summer 2015 Recursive Def of Arithmetic Expressions  Basis: a number or a variable is an expression.  If E and F are expressions then a new expression G can be formed by applying one of the three rules 1.G = E + F 2.G = E * F 3.G = ( E )

26. – 26 – CSCE 355 Summer 2015 Every Expression has equal number of left and right parenthses

27. – 27 – CSCE 355 Summer 2015 Homework 1.. 2.Prove if a complete binary tree has n leaves then it has 2n-1 nodes.

28. – 28 – CSCE 355 Summer 2015 References– Mathematical Foundations  http://en.wikipedia.org/wiki/Binary_relation http://en.wikipedia.org/wiki/Binary_relation  http://en.wikipedia.org/wiki/Relation_(mathematics)  http://en.wikipedia.org/wiki/Mathematical_proof http://en.wikipedia.org/wiki/Mathematical_proof  http://en.wikipedia.org/wiki/Proofs_from_THE_BOOK http://en.wikipedia.org/wiki/Proofs_from_THE_BOOK Extended “Proof” techniques  http://www.maths.uwa.edu.au/~berwin/humour/invalid. proofs.html http://www.maths.uwa.edu.au/~berwin/humour/invalid. proofs.html http://www.maths.uwa.edu.au/~berwin/humour/invalid. proofs.html Fair Use Books Online  http://fair-use.org/bertrand-russell/the-principles-of- mathematics/ http://fair-use.org/bertrand-russell/the-principles-of- mathematics/ http://fair-use.org/bertrand-russell/the-principles-of- mathematics/Books Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills