Discrete Mathematics Lecture 4: Sequences and Mathematical Induction, Slides of Discrete Mathematics

Download Discrete Mathematics Lecture 4: Sequences and Mathematical Induction and more Slides Discrete Mathematics in PDF only on Docsity!

Discrete Mathematics

Lecture 4: Sequences

and Mathematical Induction

Sequences

• Sequence is a set of (usually infinite number of) ordered elements: a 1 , a 2 , …, an, …
• Each individual element ak is called a term, where k is called an index
• Sequences can be computed using an explicit formula: ak = 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 Docsity.com

Exercises

• Consider the sequence defined by

an = (2n + (-1) n^ – 1) / 4 Find an alternative formula for an that uses floor notation

• Show that n! + k is divisible by k for all integers k no greater than n.
• Given any integer m greater than 1, is it possible to find a sequence of m – 1 consecutive positive integers none of which is prime?

Mathematical Induction

• Can we represent any amount greater than 3 cents as a combination of 2 and 5 cents?
• 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

Exercises

• Show that 2 2n^ – 1 is divisible by 3
• Show 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
• On the outside rim of a circular disk the integers from 1

to 30 are painted in random. Show that there must be three successive integers whose sum is at least 45

Exercises

• Show that for any integers a and b there

exist two integer u and v so that

gcd(a, b) = ua + vb

• Show that every integer greater than 1 is

divisible by a prime number

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 pre- and 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

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 Algorithms

• Division Algorithm

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