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

2. Models of Computation

3. 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

4. Prerequisites CSCE 211 Number systems, Boolean algebra, logic design,
sequential machines
Mealy machines
Moore machines

5. Prerequisites CSCE 350 MATH 374
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.

6. 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

7. Special types of Relations
Injections
Surjections
Functions

8. Properties of Relations
Property
Def
Example
Neg-Example
Reflexive
Irreflexive
symmetric
antisymmetric
asymmetric
transitive
Total
Injection
Surjection
function

9. Posets Partially Ordered Sets (POSETS) Hasse Diagram
Reflexive
Antisymmetric
Transitive
Hasse Diagram
Topological sorting

10. Equivalence relations

11. 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

12. 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

13. Theorem 1.3 used to prove Theorem 1.4
Theorem 1.3 If x >= 4 then 2x >= x2.
Theorem 1.4 If x is the sum of the squares of 4 positive integers then 2x >= x2.
Proof

14. Theorem 1.3 If x >= 4 then 2x >= x2.
f(x) = x2 / 2x.
Then what is the derivative f’ of f
Derivative of quotient??
So f’(x) =

15. 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

16. Proof by Contradiction

17. 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

18. 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

19. Example Induction Proof: Theorem 1.16

20. Number of leaves in complete tree of height h is 2h.

21. More general induction
Basis step as before
Assume S(k) for all k <= n then show S(n)

22. 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
Add new node N, the root of the new tree
Add copies of T1… Tk
Add an edge from N to the root of each T1, T2, … Tk

23. 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

24. 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
G = E + F
G = E * F
G = ( E )

25. Every Expression has equal number of left and right parenthses

26. Homework .
Prove if a complete binary tree has n leaves then it has 2n-1 nodes.

27. References– Mathematical Foundations
Extended “Proof” techniques
Fair Use Books Online
Books
Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills