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Question 5 [12 pts]: Countability. A set S is countable if either it is finite or it has the same size as N, that is, there exists a bijective function (both one-to-one and onto) from N to S. To show S is countable, you can also construct an onto function (surjection) from N to S. Here N is the set of natural numbers, N = {1,2,...}. (a) Prove that if sets A and B are countable, then AUB is also countable. (b) Prove that the set of rational numbers Q is countable. A real number is rational if it can be written as a fraction p/q, where p is an integer and q is a natural number. Hence, $Q = \{\frac{m}{n} | m \in Z, n \in N\}$ Here, Z = {..., -2, -1, 0, 1, 2,...} is the set of integers. [Hint: Example 4.15 in the Sipser book argues about countability of positive rational numbers. You can assume Z is countable if needed.] (c) Using Parts (a) and (b), prove that the set of irrational numbers is an uncountable set. [Hint: You can use the fact that the set of real numbers is uncountable.]

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As a part of a medical research study, a researcher exposes two groups of participants to either an actual painkiller or a sugar pill. The participants feel their pain being eliminated even with the sugar pill. In this case, the sugar pill is known as the O confederate. O random sample. O placebo. O independent variable.

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Consider the following dataset that shows the scores of students in standardized exams on reading, writing as well as Math and Science. Determine the Canonical Correlation Coefficient between {read, write} and {Math, Science} scores. Num read write math science 1 54.8 64.5 44.5 52.6 2 52.1 51.5 57.9 60.7 3 44.2 35.9 43.6 47.1 4 62.7 59.3 56.5 55.3 5 52.1 59.3 58.1 47.1 6 46.9 44.3 48.7 53.1 7 46.9 61.9 46.2 60.7 8 49.5 54.1 38.7 49.8 9 44.2 54.1 47.1 58 10 38.4 37.2 33.4 30.9 11 44.2 33.3 53.9 58 12 46.9 61.9 53 52.6 13 49.5 51.5 55.5 44.4

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Find the curvature. k(x)=(y)/((36cos(3x))/((1+144sin^(2)(3x))^((3)/(2))))=4cos(3x) Find the curvature y=4cos3x 36cos3x kx)= 1+144 sin23x

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Give the main product of the following reaction, involved in the metabolism of formaldehyde (which is poisonous): OH HO O OH OH + O H H Draw the molecule on the canvas by choosing buttons from the Tools (for bonds and charges), Atoms, and Templates toolbars.

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c) During the lecture hour, you have been explained about an example of modelling the environmental impacts for one kilogram of laundry to be washed using Monte Carlo simulation. Explain another example of scenario where you can practice Monte Carlo simulation.

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Write a program to compute the absolute and relative errors in Stirling's approximation $n! \approx \sqrt{2\pi n}(n/e)^n$ for $n = 1, 2, \dots, 10$. Does the absolute error grow or shrink as $n$ increases? Does the relative error grow or shrink as $n$ increases? Is the result affected when using double precision instead of single precision?

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Question1. (20 points) Considering algorithms implemented on a computer satisfying: (1) For all real numbers $x$, there exists $\epsilon$ with $|\epsilon| \le \epsilon_{machine}$ such that $fl(x) = x(1 + \epsilon)$. (2) For all floating point numbers $x$ and $y$, there exists $\epsilon$ with $|\epsilon| \le \epsilon_{machine}$ such that $x \odot y = (x * y)(1 + \epsilon)$, where $*) and \(\odot$ represent exact arithmetic operations $(+, -, \times, \text{or} \div)$ and the corresponding floating point analogue, respectively. Judge the backward stability of the following algorithms: (a) For real number $x$, compute $x^2$ as $x \odot x$. (b) For real number $x$, compute $x + 1$ as $x \oplus 1$.

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Question 2 Your task is to write a Python program that takes an integer as input and prints the squares of integers up to and including that number. A sample session follows: Program prompts: What number do we have today? The user responds: 4 The program prints: Here is your series: 1 4 9 16

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