Carnegie Mellon University in Qatar
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Carnegie Mellon University in Qatar (جامعة كارنيجي ميلون في قطر) ** School Info Carnegie Mellon University in Qatar (جامعة كارنيجي ميلون في قطر) has 11 departments in Course Hero with 104 documents and 8 answered questions.

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Carnegie Mellon University in Qatar Documents (104)

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  • 4 pages Homework6-AssignedSolutions.pdf
    • Homework6-AssignedSolutions.pdf
    • Carnegie Mellon University in Qatar
    • Concepts of Math
    • MATH 21-127 - Spring 2014
    • Homework 6 Solutions to Assigned Problems 1. Define on N N by (a, b) (c, d) a + b = c + d. Claim: is an equivalence relation on N N. Proof: We show is an equivalence relation by showing it is reflexive, symmetric and transitive. R: Let (a, b) N N. Since a
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    Homework6-AssignedSolutions.pdf

  • 3 pages Homework9-AssignedSolutions.pdf
    • Homework9-AssignedSolutions.pdf
    • Carnegie Mellon University in Qatar
    • Concepts of Math
    • MATH 21-127 - Spring 2014
    • Homework 9 Solutions to Assigned Problems 1. (a) I is uncountable. Since R is uncountable, Q is countable, and R = I Q it must be the case that I is uncountable. (b) A is countably infinite. Each circle is determined by its center and radius. There is a d
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    Homework9-AssignedSolutions.pdf

  • 4 pages Homework10-Assigned.pdf
    • Homework10-Assigned.pdf
    • Carnegie Mellon University in Qatar
    • Concepts of Math
    • MATH 21-127 - Spring 2014
    • Homework 10 Solutions to Assigned Problems 1. Let U be the set of all 5-card hands that have exactly two kings and exactly one heart. Partition U by cfw_A, B where A is the set of hands with exactly two kings and one heart that DOES contain the king of he
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    Homework10-Assigned.pdf

  • 2 pages Homework2-AssignedSolutions.pdf
    • Homework2-AssignedSolutions.pdf
    • Carnegie Mellon University in Qatar
    • Concepts of Math
    • MATH 21-127 - Spring 2014
    • Homework 2 Solutions to Assigned Problems 1. Claim: For sets A and B, if P(A) = P(B) then A = B. Proof: Suppose P(A) = P(B). Let a A be arbitrary. Then cfw_a A and hence cfw_a P(A) by definition of a power set. Then cfw_a P(B) since we assumed P(A) = P(B)
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    Homework2-AssignedSolutions.pdf

  • 3 pages Homework5-Assigned Solutions.pdf
    • Homework5-Assigned Solutions.pdf
    • Carnegie Mellon University in Qatar
    • Concepts of Math
    • MATH 21-127 - Spring 2014
    • Homework 5 Solutions to Assigned Problems 1. (a) Claim: n N. 1 fn+1 fn 2 Proof: Let P (n) be the proposition 1 Base: n = 1. fn+1 fn = f2 f1 fn+1 fn 2. We proceed by induction on n N. = 11 . Clearly 1 1 2. Thus P (1) holds. f IH: Let k N and assume P (k
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    Homework5-Assigned Solutions.pdf

  • 3 pages Homework4-AssignedSolutions.pdf
    • Homework4-AssignedSolutions.pdf
    • Carnegie Mellon University in Qatar
    • Concepts of Math
    • MATH 21-127 - Spring 2014
    • Homework 4 Solutions to Assigned Problems 1. Claim: 3n 2n+1 for all natural numbers n 2. Proof: We proceed by induction. For the base case of n = 2 we can see that 32 = 9 23 = 8, as desired. Now, for the inductive step, assume 3k 2k+1 for some k 2. Then w
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    Homework4-AssignedSolutions.pdf

  • 3 pages Homework8-AssignedSolutions.pdf
    • Homework8-AssignedSolutions.pdf
    • Carnegie Mellon University in Qatar
    • Concepts of Math
    • MATH 21-127 - Spring 2014
    • Homework 8 Solutions to Assigned Problems 1. Let f : A B be a function, X1 , X2 A, and Y1 , Y2 B. (a) Claim: Imf (X1 ) Imf (X2 ) Imf (X1 X2 ) Proof: Let b Imf (X1 ) Imf (X2 ). Then b Imf (X1 ) and b / Imf (X2 ). Since b Imf (X1 ) there exists x X1 such t
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    Homework8-AssignedSolutions.pdf

  • 2 pages Homework3-AssignedSolutions.pdf
    • Homework3-AssignedSolutions.pdf
    • Carnegie Mellon University in Qatar
    • Concepts of Math
    • MATH 21-127 - Spring 2014
    • Homework 3 Solutions to Assigned Problems 1. Below is the statement P : x, y Z. z Z. (x y > z x y) Claim: P holds Proof: Let x, y Z be arbitrary. Consider z = 0 Z. And, if x y > 0 then we have x > y as desired. Thus z = 0 satisfies the claim. (Additional
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    Homework3-AssignedSolutions.pdf

  • 5 pages Homework7-AssignedSolutions.pdf
    • Homework7-AssignedSolutions.pdf
    • Carnegie Mellon University in Qatar
    • Concepts of Math
    • MATH 21-127 - Spring 2014
    • Homework 7 Solutions to Assigned Problems 1. Claim: If p is a prime number with p > 3 then p2 1 (mod 24). Proof: We break the proof into 3 claims, with the third claim completing the proof: Claim 1: p2 1 (mod 8) Proof: Since p is prime and p > 3, p must
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    Homework7-AssignedSolutions.pdf

  • 5 pages recitation11sol.pdf
    • recitation11sol.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • 15-251: Great Theoretical Ideas In Computer Science Recitation 11 : Probability and Randomized Algorithms Solutions Lecture Review Let X, Y : R be random variables. The expectation of X is defined to be X X x Pr[X = x] = Pr[`]X(`). E[X] = xrange(X) ` Ex
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    recitation11sol.pdf

  • 77 pages lec5.pdf
    • lec5.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • 15-251: Great Theoretical Ideas in Computer Science Lecture 5 Turings Legacy What is computation? What is an algorithm? How can we mathematically define them? Quick Recap Mathematical definition of a (computational) problem: Input / output function: Langu
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    lec5.pdf

  • 3 pages recitation2sol.pdf
    • recitation2sol.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • 15-251: Great Theoretical Ideas In Computer Science Recitation 2 Solutions Regular Announcements Homework Solution Sessions - Friday 5p-6p, Saturday 2p-3p in GHC 4301 Small Groups this weekend Come to us if you had difficulties on Homework 1 Definition
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    recitation2sol.pdf

  • 3 pages recitation14sol.pdf
    • recitation14sol.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • 15-251: Great Theoretical Ideas In Computer Science Recitation 13 Solutions Announcements Homework Solution Sessions - Friday 5p-6p, Saturday 2p-3p in GHC 4301 Small groups sessions this weekend - sign up if youd like to review material for the final De
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    recitation14sol.pdf

  • 3 pages induction-pitfalls.pdf
    • induction-pitfalls.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • 15-251: Great Theoretical Ideas Dos and Donts in Inductive Proofs Consider the problem of proving that n 0, 1 + 2 + . . . + n = Define the statement Sn = 1 + 2 + . . . + n = 1 n(n+1) . 2 n(n+1) 2 by induction. We want to prove n 0, Sn . An Inductive Proof
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    induction-pitfalls.pdf

  • 4 pages homework12.pdf
    • homework12.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • CMU 15-251 Spring 2017 Homework 12 Writing session on Wednesday May 3 1. (SOLO) (a) First, review the RSA cryptosystem. Well encode and decode a simple message using RSA. The message is allowed to be any string containing only lower case letters and space
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    homework12.pdf

  • 2 pages recitation5.pdf
    • recitation5.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • 15-251: Great Theoretical Ideas In Computer Science Recitation 5 The running time of an algorithm A is a function TA : N N dened by TA (n) = maxIS cfw_number of steps A takes on I, where S is the set of instances I of size n. For f, g : N+ R+ , we say f
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    recitation5.pdf

  • 5 pages recitation3sol.pdf
    • recitation3sol.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • 15-251: Great Theoretical Ideas In Computer Science Recitation 3 Solutions Announcements Be sure to take advantage of the following resources : Homework Solution Sessions - Friday 5p-6p, Saturday 2p-3p in GHC 4301 Conceptual office hours (no HW help) -
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    recitation3sol.pdf

  • 3 pages homework3.pdf
    • homework3.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • CMU 15-251 Spring 2017 Homework 3 Writing session on Wednesday February 8 1. (SOLO) (a) Let L = cfw_wwR | w , where = cfw_a, b. Draw the state diagram of a TM deciding L. As usual, you can use any finite tape alphabet containing and t; just be sure to spe
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    homework3.pdf

  • 6 pages recitation9sol.pdf
    • recitation9sol.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • 15-251: Great Theoretical Ideas In Computer Science Recitation 9 : P and NP Solutions News Post NP and NP-completeness Review Session Saturday, March 25 from 11am-12:30pm in GHC 4301 Solution Sessions for both HW6 and HW7 Friday 5-7pm in GHC 4102 and Sa
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    recitation9sol.pdf

  • 2 pages homework2.pdf
    • homework2.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • CMU 15-251 Spring 2017 Homework 2 Writing session on Wednesday February 1 1. (SOLO) Which of the following languages are regular? If you think a language is regular, draw the state diagram of a DFA that recognizes the language. You do not have to prove th
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    homework2.pdf

  • 3 pages homework8.pdf
    • homework8.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • CMU 15-251 Spring 2017 Homework 8 Writing session on Wednesday March 29 Note: If a question asks you to show that a language is in NP, you must closely follow the formal definition of NP in your proofs. For this homework, you may assume that the following
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    homework8.pdf

  • 6 pages recitation1sol.pdf
    • recitation1sol.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • 15-251: Great Theoretical Ideas In Computer Science Recitation 1 Solutions Announcements Conceptual office hours are today, 6-8 PM in Gates 5 Carrel 1 (the double carrel). Use this time to clarify concepts from lecture. Check the website for times and l
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    recitation1sol.pdf

  • 3 pages homework9.pdf
    • homework9.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • CMU 15-251 Spring 2017 Homework 9 Writing session on Wednesday April 5 For this homework, you may assume that the following problems are NP-complete: CIRCUIT-SAT: Given a Boolean circuit with AND, OR, and NOT gates, is there an assignment to the input var
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    homework9.pdf

  • 5 pages recitation7sol.pdf
    • recitation7sol.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • 15-251: Great Theoretical Ideas In Computer Science Recitation 7 Solutions A A A A matching in G is a subset of Gs edges which share no vertices. maximal matching is one which isnt a subset of any other matching. maximum matching is a matching which is a
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    recitation7sol.pdf

  • 2 pages homework11.pdf
    • homework11.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • CMU 15-251 Spring 2017 Homework 11 Writing session on Wednesday April 26 1. (SOLO) Consider a variant of the minimum cut problem where each edge has a cost, and our goal is to output a cut with minimum total cost (the cost of a cut is the sum of the costs
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    homework11.pdf

  • 4 pages recitation4sol.pdf
    • recitation4sol.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • 15-251: Great Theoretical Ideas In Computer Science Recitation 4 Solutions Announcements Be sure to take advantage of the following resources : Homework Solution Sessions - Friday 5p-6p, Saturday 2p-3p in GHC 4301 Conceptual oce hours (no HW help) - Fri
    • Register Now

    recitation4sol.pdf

  • 2 pages recitation4.pdf
    • recitation4.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • 15-251: Great Theoretical Ideas In Computer Science Recitation 4 Announcements Be sure to take advantage of the following resources : Homework Solution Sessions - Friday 5p-6p, Saturday 2p-3p in GHC 4301 Conceptual oce hours (no HW help) - Friday 6p-8p
    • Register Now

    recitation4.pdf

  • 2 pages homework10.pdf
    • homework10.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • CMU 15-251 Spring 2017 Homework 10 Writing session on Wednesday April 12 1. (SOLO) In the 1-d Battleship Problem, the input is an array B[1 . . . n] consisting of n/4 1s and 3n/4 0s. The task is to output an index i such that B[i] = 1. Write a randomized
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    homework10.pdf

  • 4 pages final-practice.pdf
    • final-practice.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • CMU 15-251 Spring 2017 Final Exam Practice Problems 1 Medium to Long Answer 1. Consider the following two computational problems: Given as input a number A, the output is a number B such that B 3 = A. If no such number exists, the output is False. Given
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    final-practice.pdf

  • 2 pages homework7.pdf
    • homework7.pdf
    • Carnegie Mellon University in Qatar
    • Great Theoretical Ideas in Computer Science
    • CS 15-251 - Spring 2017
    • CMU 15-251 Spring 2017 Homework 7 Writing session on Wednesday March 22 1. (SOLO) A language L cfw_0, 1 is called skinny if there is some constant k 0 such that for all n N, we have |L cfw_0, 1n | nk . (a) Show that any skinny language can be computed by
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    homework7.pdf

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