Lecture Notes - jan.ucc.nau.edu
... Proof by Contradiction: Example • Theorem: There exists an infinite number of prime numbers. • Proof (courtesy of Euclid): 1. Assume that there are a finite number of primes. 2. Then there is a largest prime, p. Consider the number q = (2x3x5x7x...xp)+1. q is one more than the product of all primes ...
... Proof by Contradiction: Example • Theorem: There exists an infinite number of prime numbers. • Proof (courtesy of Euclid): 1. Assume that there are a finite number of primes. 2. Then there is a largest prime, p. Consider the number q = (2x3x5x7x...xp)+1. q is one more than the product of all primes ...
On a strong law of large numbers for monotone measures
... (II) We say that Xn is almost everywhere convergent to X and write Xn −→ X if there exists a υ-nullset N ∈ F such that ∀ω ∈ N c : |Xn (ω) − X (ω)| → 0 ...
... (II) We say that Xn is almost everywhere convergent to X and write Xn −→ X if there exists a υ-nullset N ∈ F such that ∀ω ∈ N c : |Xn (ω) − X (ω)| → 0 ...
Lecture 18: recursion
... Suppose we can sort k numbers in Tk steps. Then we can sort 2k numbers in 2Tk + 2k steps. Therefore, T2k = 2Tk + 2k. If we solve this recurrence (which we will do later), then we see that T2n ≈ n log2 n. This is significantly faster than bubble sort! ...
... Suppose we can sort k numbers in Tk steps. Then we can sort 2k numbers in 2Tk + 2k steps. Therefore, T2k = 2Tk + 2k. If we solve this recurrence (which we will do later), then we see that T2n ≈ n log2 n. This is significantly faster than bubble sort! ...
Chapter 5
... “permutations” to “combinations” order must be disregarded. In how many ways can we permute r objects? The answer is r!. Therefore we must divide our formula from (3) and (4) by r!. ...
... “permutations” to “combinations” order must be disregarded. In how many ways can we permute r objects? The answer is r!. Therefore we must divide our formula from (3) and (4) by r!. ...
DS Lecture 6
... Let Q(x, y) be the statement x+y=x−y where the domain for x and y is the set of all real numbers. Determine the truth value of: (a) Q(5,−2). (b) Q(4.7, 0). (c) Determine the set of all pairs of numbers, x and y, such that Q(x, y) is true. Solution: (a) Q(5,−2) says that 5 + (−2) = 5 − (−2), or 3= 7, ...
... Let Q(x, y) be the statement x+y=x−y where the domain for x and y is the set of all real numbers. Determine the truth value of: (a) Q(5,−2). (b) Q(4.7, 0). (c) Determine the set of all pairs of numbers, x and y, such that Q(x, y) is true. Solution: (a) Q(5,−2) says that 5 + (−2) = 5 − (−2), or 3= 7, ...
Elementary mathematics
Elementary mathematics consists of mathematics topics frequently taught at the primary or secondary school levels. The most basic topics in elementary mathematics are arithmetic and geometry. Beginning in the last decades of the 20th century, there has been an increased emphasis on problem solving. Elementary mathematics is used in everyday life in such activities as making change, cooking, buying and selling stock, and gambling. It is also an essential first step on the path to understanding science.In secondary school, the main topics in elementary mathematics are algebra and trigonometry. Calculus, even though it is often taught to advanced secondary school students, is usually considered college level mathematics.