CS322

1. Week 15 - Wednesday CS322

2. Last time • What did we talk about last time? • Review first third of course

3. Questions?

4. Logical warmup • Consider the following shape to the right: • Now, consider the next shape, made up of pieces of exactly the same size: • We have created space out of nowhere! • How is this possible?

5. Indirect Proofs

6. Proof by contradiction • In a proof by contradiction, you begin by assuming the negation of the conclusion • Then, you show that doing so leads to a logical impossibility • Thus, the assumption must be false and the conclusion true

7. Contradiction formatting • A proof by contradiction is different from a direct proof because you are trying to get to a point where things don't make sense • You should always mark such proofs clearly • Start your proof with the words Proof by contradiction • Write Negation of conclusion as the justification for the negated conclusion • Clearly mark the line when you have both p and ~p as a contradiction • Finally, state the conclusion with its justification as the contradiction found before

8. Practice • Use a proof by contradiction to prove the following: • For all integers n, if n2 is odd then n is odd • For all prime numbers a, b, and c, a2 + b2≠ c2

9. Sequences and Induction

10. Sequences • Mathematical sequences can be represented in expanded form or with explicit formulas • Examples: • 2, 5, 10, 17, 26, … • ai = i2 + 1, i ≥ 1 • Summation notation is used to describe a summation of some part of a series • Product notation is used to describe a product of some part of a series

11. Proof by mathematical induction • To prove a statement of the following form: • n  Z, where n a, property P(n) is true • Use the following steps: • Basis Step: Show that the property is true for P(a) • Induction Step: • Suppose that the property is true for some n = k, where k  Z, k a • Now, show that, with that assumption, the property is also true for k + 1

12. Practice • Write the following in closed form: • Use mathematical induction to prove: • For all integers n ≥ 1, 2 + 4 + 6+· · ·+2n = n2 + n

13. Recursion • Using recursive definitions to generate sequences • Writing a recursive definition based on a sequence • Using mathematical induction to show that a recursive definition and an explicit definition are equivalent

14. Solving recursion by iteration • Expand the recursion repeatedly without combining like terms • Find a pattern in the expansions • When appropriate, employ formulas to simplify the pattern • Geometric series: 1 + r + r2+ … + rn = (rn+1 – 1)/(r – 1) • Arithmetic series: 1 + 2 + 3 + … + n = n(n+ 1)/2

15. Practice • Use the method of iteration to find an explicit formula for the following recursively defined sequence: • dk= 2dk−1 + 3, for all integers k ≥ 2 • d1= 2 • Use a proof by induction to show that your explicit formula is correct

16. Solving second order linear homogeneous recurrence relations with constant coefficients • To solve sequence ak = Aak-1 + Bak-2 • Find its characteristic equation t2 – At – B = 0 • If the equation has two distinct roots r and s • Substitute a0 and a1 into an = Crn + Dsn to find C and D • If the equation has a single root r • Substitute a0 and a1 into an = Crn + Dnrn to find C and D

17. Practice • Find an explicit formula for the following: • rk= 2rk-1− rk-2, for all integers k ≥ 2 • r0= 1 • r1= 4

18. Set Theory

19. Set theory basics • Defining finite and infinite sets • Definitions of: • Subset • Proper subset • Set equality • Set operations: • Union • Intersection • Difference • Complement • The empty set • Partitions • Cartesian product

20. Set theory proofs • Proving a subset relation • Element method: Assume an element is in one set and show that it must be in the other set • Algebraic laws of set theory: Using the algebraic laws of set theory (given on the next slide), we can show that two sets are equal • Disproving a universal statement requires a counterexample with specific sets

21. Laws of set theory

22. Russell's paradox • It is possible to give a description for a set which describes a set that does not actually exist • For a well-defined set, we should be able to say whether or not a given element is or is not a member • If we can find an element that must be in a specific set and must not be in a specific set, that set is not well defined

23. Functions • Definitions • Domain • Co-domain • Range • Inverse image • Arrow diagrams • Poorly defined functions • Function equality

24. Inverses • One-to-one (injective) functions • Onto (surjective) functions • If a function F: X Yis both one-to-one and onto (bijective), then there is an inverse function F-1: Y X such that: • F-1(y) = x F(x) = y, for all x X and y  Y

25. Cardinality • Pigeonhole principle: • If n pigeons fly into m pigeonholes, where n > m, then there is at least one pigeonhole with two or more pigeons in it • Cardinality is the number of things in a set • It is reflexive, symmetric, and transitive • Two sets have the same cardinality if a bijective function maps every element in one to an element in the other • Any set with the same cardinality as positive integers is called countably infinite

26. Practice • Consider the set of integer complex numbers, defined as numbers a + bi, where a, bZ and i is • Prove that the set of integer complex numbers is countable

27. Relations

28. Relations • Relations are generalizations of functions • In a relation (unlike functions), an element from one set can be related to any number (from zero up to infinity) of other elements • We can define any binary relation between sets A and B as a subset of A x B • If x is related to y by relation R, we write x R y • All relations have inverses (just reverse the order of the ordered pairs)

29. Properties of relations • For relation R on set A • R is reflexiveiff for all x A, (x, x)  R • R is symmetriciff for all x, y A, if (x, y)  R then (y, x)  R • R is transitiveiff for all x, y, z A, if (x, y)  R and (y, z)  R then (x, z)  R • R is antisymmetriciff for all a and b in A, if a R b and b R a, then a = b • The transitive closure of R called Rt satisfies the following properties: • Rt is transitive • R Rt • If S is any other transitive relation that contains R, then Rt  S

30. Equivalence relations and partial orders • Let A be partitioned by relation R • R is reflexive, symmetric, and transitiveiff it induces a partition on A • We call a relation with these three properties an equivalence relation • Example: congruence mod 3 • If R is reflexive, antisymmetric, and transitive, it is called a partial order • Example: less than or equal

31. Practice • Prove that the subset relationship is a partial order • Consider the relation x R y, where R is defined over the set of all people • x R y↔ x lives in the same house as y • Is R an equivalence relation? Prove it.

32. Counting and Probability

33. Probability definitions • A sample space is the set of all possible outcomes • An event is a subset of the sample space • Formula for equally likely probabilities: • Let S be a finite sample space in which all outcomes are equally likely and E is an event in S • Let N(X) be the number of elements in set X • Many people use the notation |X| instead • The probability of E is P(E) = N(E)/N(S)

34. Multiplication rule • If an operation has k steps such that • Step 1 can be performed in n1 ways • Step 2 can be performed in n2 ways … • Step k can be performed in nk ways • Then, the entire operation can be performed in n1n2 … nk ways • This rule only applies when each step always takes the same number of ways • If each step does not take the same number of ways, you may need to draw a possibility tree

35. Addition and inclusion/exclusion rules • If a finite set Aequals the union of k distinct mutually disjoint subsets A1, A2, … Ak, then: N(A) = N(A1) + N(A2) + … + N(Ak) • If A, B, C are any finite sets, then: N(A B) = N(A) + N(B) – N(A  B) • And: N(A B  C) = N(A) + N(B) + N(C) – N(A  B) – N(A  C) – N(B  C) + N(A  B  C)

36. Counting guide • This is a quick reminder of all the different ways you can count k things drawn from a total of n things: • Recall that P(n,k) = n!/(n – k)! • And = n!/((n – k)!k!)

37. Binomial theorem • The binomial theorem states: • You can easily compute these coefficients using Pascal's triangle for small values of n

38. Probability axioms • Let A and B be events in the sample space S • 0 ≤ P(A) ≤ 1 • P() = 0 and P(S) = 1 • If A  B = , then P(A  B) = P(A) + P(B) • It is clear then that P(Ac) = 1 – P(A) • More generally, P(A  B) = P(A) + P(B) – P(A  B)

39. Expected value • Expected value is one of the most important concepts in probability, especially if you want to gamble • The expected value is simply the sum of all events, weighted by their probabilities • If you have n outcomes with real number values a1, a2, a3, … an, each of which has probability p1, p2, p3, … pn, then the expected value is:

40. Conditional probability • Given that some event A has happened, the probability that some event B will happen is called conditional probability • This probability is:

41. Quiz

42. Upcoming

43. Next time… • Review third third of the course

44. Reminders • Review chapters 10 – 12 and notes on grammars and automata