Discrete Mathematics Lecture 4 Harper Langston New York University
Sequences Sequence is a set of (usually infinite number of) ordered elements: a 1, a 2,, a n, Each individual element a k is called a term, where k is called an index Sequences can be computed using an explicit formula: a k = k * (k + 1) for k > 1 Alternate sign sequences Finding an explicit formula given initial terms of the sequence: 1, -1/4, 1/9, -1/16, 1/25, -1/36, Sequence is (most often) represented in a computer program as a single-dimensional array
Sequence Operations Summation: Σ, expanded form, limits (lower, upper) of summation, dummy index Change of index inside summation Product: Π, expanded form, limits (lower, upper) of product, dummy index Factorial: n!, n! = n * (n 1)!
Review Mathematical Induction Principle of Mathematical Induction: Let P(n) be a predicate that is defined for integers n and let a be some integer. If the following two premises are true: P(a) is a true k a, P(k) P(k + 1) then the following conclusion is true as well P(n) is true for all n a
Applications of Mathematical Induction Show that 1 + 2 + + n = n * (n + 1) / 2 (Prove on board) Sum of geometric series: r 0 + r 1 + + r n = (r n+1 1) / (r 1) (Prove on board)
Examples that Can be Proved with Mathematical Induction Show that 2 2n 1 is divisible by 3 (in book) Show (on board) that for n > 2: 2n + 1 < 2 n Show that x n y n is divisible by x y Show that n 3 n is divisible by 6 (similar to book problem)
Strong Mathematical Induction Utilization of predicates P(a), P(a + 1),, P(n) to show P(n + 1). Variation of normal M.I., but basis may contain several proofs and in assumption, truth assumed for all values through from base to k. Examples: Any integer greater than 1 is divisible by a prime Existence and Uniqueness of binary integer representation (Read in book)
Well-Ordering Principle Well-ordering principle for integers: a set of integers that are bounded from below (all elements are greater than a fixed integer) contains a least element Example: Existence of quotient-remainder representation of an integer n against integer d
Stepping back: Algorithms Last lecture we talked about some elementary number theory such as the Quotient Remainder Theorem (pg 157) Algorithm is step-by-step method for performing some action (such as finding remainder in Q.R.) Cost of statements execution Simple statements Conditional statements Iterative statements
Example: Division Algorithm Input: integers a and d Output: quotient q and remainder r Body: r = a; q = 0; while (r >= d) r = r d; q = q + 1; end while
Greatest Common Divisor The greatest common divisor of two integers a and b is another integer d with the following two properties: d a and d b if c a and c b, then c d Lemma 1: gcd(r, 0) = r Lemma 2: if a = b * q + r, then gcd(a, b) = gcd(b, r)
Euclidean Algorithm Input: integers a and b (a>b>=0) Output: greatest common divisor gcd Body: r = b; while (b > 0) r = a mod b; a = b; b = r; end while gcd = a;
Exercise Least common multiple: lcm Prove that for all positive integers a and b, gcd(a, b) = lcm(a, b) iff a = b
Correctness of Algorithms Assertions Pre-condition is a predicate describing initial state before an algorithm is executed Post-condition is a predicate describing final state after an algorithm is executed Loop guard Loop is defined as correct with respect to its preand post- conditions, if whenever the algorithm variables satisfy the pre-conditions and the loop is executed, then the algorithm satisfies the post-conditions as well
Loop Invariant Theorem Let a while loop with guard G be given together with its pre- and post- conditions. Let predicate I(n) describing loop invariant be given. If the following 4 properties hold, then the loop is correct: Basis Property: I(0) is true before the first iteration of the loop Inductive Property: If G and I(k) is true, then I(k + 1) is true Eventual Falsity of the Guard: After finite number of iterations, G becomes false Correctness of the Post-condition: If N is the least number of iterations after which G becomes false and I(N) is true, then post-conditions are true as well
Correctness of Some Algorithms Product Algorithm: pre-conditions: m 0, i = 0, product = 0 while (i < m) { product += x; i++; } post-condition: product = m * x
Correctness of Some Division Algorithm Algorithms pre-conditions: a 0, d > 0, r = a, q = 0 while (r d) { r -= d; } q++; post-conditions: a = q * d + r, 0 r < d
Correctness of Some Euclidean Algorithm Algorithms pre-conditions: a > b 0, r = b while (b > 0) { } r = a mod b; a = b; b = r; post-condition: a = gcd(a, b)
Basics of Set Theory Set and element are undefined notions in the set theory and are taken for granted Set notation: {1, 2, 3}, {{1, 2}, {3}, {1, 2, 3}}, {1, 2, 3, },, {x R -3 < x < 6} Set A is called a subset of set B, written as A B, when x, x A x B. What is negation? A is a proper subset of B, when A is a subset of B and x B and x A Visual representation of the sets Distinction between and
Set Operations Set a equals set B, iff every element of set A is in set B and vice versa. (A = B A B /\ B A) Proof technique for showing sets equality (example) Union of two sets is a set of all elements that belong to at least one of the sets (notation on board) Intersection of two sets is a set of all elements that belong to both sets (notation on board) Difference of two sets is a set of elements in one set, but not the other (notation on board) Complement of a set is a difference between universal set and a given set (notation on board) Examples
Empty Set S = {x R, x 2 = -1} X = {1, 3}, Y = {2, 4}, C = X Y (X and Y are disjoint) Empty set has no elements Empty set is a subset of any set There is exactly one empty set Properties of empty set: A = A, A = A A c =, A A c = U U c =, c = U
Set Partitioning Two sets are called disjoint if they have no elements in common Theorem: A B and B are disjoint A collection of sets A 1, A 2,, A n is called mutually disjoint when any pair of sets from this collection is disjoint A collection of non-empty sets {A 1, A 2,, A n } is called a partition of a set A when the union of these sets is A and this collection consists of mutually disjoint sets
Power Set Power set of A is the set of all subsets of A Example on board Theorem: if A B, then P(A) P(B) Theorem: If set X has n elements, then P(X) has 2 n elements (proof in Section 5.3 will show if have time)
Cartesian Products Ordered n-tuple is a set of ordered n elements. Equality of n-tuples Cartesian product of n sets is a set of n- tuples, where each element in the n-tuple belongs to the respective set participating in the product
Set Properties Inclusion of Intersection: A B A and A B B Inclusion in Union: A A B and B A B Transitivity of Inclusion: (A B B C) A C Set Definitions: x X Y x X y Y x X Y x X y Y x X Y x X y Y x X c x X (x, y) X Y x X y Y
Set Identities Commutative Laws: A B = A B and A B = B A Associative Laws: (A B) C = A (B C) and (A B) C = A (B C) Distributive Laws: A (B C) = (A B) (A C) and A (B C) = (A B) (A C) Intersection and Union with universal set: A U = A and A U = U Double Complement Law: (A c ) c = A Idempotent Laws: A A = A and A A = A De Morgan s Laws: (A B) c = A c B c and (A B) c = A c B c Absorption Laws: A (A B) = A and A (A B) = A Alternate Representation for Difference: A B = A B c Intersection and Union with a subset: if A B, then A B = A and A B = B
Proving Equality First show that one set is a subset of another (what we did with examples before) To show this, choose an arbitrary particular element as with direct proofs (call it x), and show that if x is in A then x is in B to show that A is a subset of B Example (step through all cases)
Disproofs, Counterexamples and Algebraic Proofs Is is true that (A B) (B C) = A C? (No via counterexample) Show that (A B) C = (A C) (B C) (Can do with an algebraic proof, slightly different)
Boolean Algebra A Boolean Algebra is a set of elements together with two operations denoted as + and * and satisfying the following properties: Commutative: a + b = b + a, a * b = b * a Associative: (a + b) + c = a + (b + c), (a * b) *c = a * (b * c) Distributive: a + (b * c) = (a + b) * (a + c), a * (b + c) = (a * b) + (a * c) Identity: a + 0 = a, a * 1 = a for some distinct unique 0 and 1 Complement: a + ã = 1, a * ã = 0
Russell s Paradox Set of all integers, set of all abstract ideas Consider S = {A, A is a set and A A} Is S an element of S? Barber puzzle: a male barber shaves all those men who do not shave themselves. Does the barber shave himself? Consider S = {A U, A A}. Is S S? Godel: No way to rigorously prove that mathematics is free of contradictions. ( This statement is not provable is true but not provable) (consistency of an axiomatic system is not provable within that system)
Halting Problem There is no computer algorithm that will accept any algorithm X and data set D as input and then will output halts or loops forever to indicate whether X terminates in a finite number of steps when X is run with data set D. Proof is by contradiction