Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter C.1, Problem 12E
Program Plan Intro
To prove inequality
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Prove that it is logically equal using common equivalent proposition.
1. ¬[p v (¬p ˄ q) ≡ (¬p ˄ ¬q)
2. p → (q ˄ ¬r)
Prove or disprove that for any x ∈ N, x(x+1)/2 ∈ N (where N = {0, 1, 2, 3, ….}
Prove f(n) = n3 - (n2 log2 n) + 2n
f(n) ∈ O(n3)
by giving the constants (c, n0) and arguing that your constants hold as n goes to infinity by either "chaining up", "chaining down", carefully treating the inequality as an equality and doing some algebra, or even weak induction.
Also, what is the minimum val4ue for c (in the definition of O() ) forj this problem and is it an exclusive or inclusive minimum/bound.
Chapter C Solutions
Introduction to Algorithms
Ch. C.1 - Prob. 1ECh. C.1 - Prob. 2ECh. C.1 - Prob. 3ECh. C.1 - Prob. 4ECh. C.1 - Prob. 5ECh. C.1 - Prob. 6ECh. C.1 - Prob. 7ECh. C.1 - Prob. 8ECh. C.1 - Prob. 9ECh. C.1 - Prob. 10E
Ch. C.1 - Prob. 11ECh. C.1 - Prob. 12ECh. C.1 - Prob. 13ECh. C.1 - Prob. 14ECh. C.1 - Prob. 15ECh. C.2 - Prob. 1ECh. C.2 - Prob. 2ECh. C.2 - Prob. 3ECh. C.2 - Prob. 4ECh. C.2 - Prob. 5ECh. C.2 - Prob. 6ECh. C.2 - Prob. 7ECh. C.2 - Prob. 8ECh. C.2 - Prob. 9ECh. C.2 - Prob. 10ECh. C.3 - Prob. 1ECh. C.3 - Prob. 2ECh. C.3 - Prob. 3ECh. C.3 - Prob. 4ECh. C.3 - Prob. 5ECh. C.3 - Prob. 6ECh. C.3 - Prob. 7ECh. C.3 - Prob. 8ECh. C.3 - Prob. 9ECh. C.3 - Prob. 10ECh. C.4 - Prob. 1ECh. C.4 - Prob. 2ECh. C.4 - Prob. 3ECh. C.4 - Prob. 4ECh. C.4 - Prob. 5ECh. C.4 - Prob. 6ECh. C.4 - Prob. 7ECh. C.4 - Prob. 8ECh. C.4 - Prob. 9ECh. C.5 - Prob. 1ECh. C.5 - Prob. 2ECh. C.5 - Prob. 3ECh. C.5 - Prob. 4ECh. C.5 - Prob. 5ECh. C.5 - Prob. 6ECh. C.5 - Prob. 7ECh. C - Prob. 1P
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- Prove the completeness axiom directly from the Nested Interval Theoremarrow_forwardAssume that A is reducible to B. Show that if B is solvable in polynomial time, then A is also solvable in polynomial time.arrow_forwardShow that ¬ (p ∨ (¬p ∧ q)) and ¬p ∧ ¬q are logically equivalent by developing a series of logical equivalences.arrow_forward
- Prove or disprove each of the following Θ(log n)^k = O(n) where k is any constant larger than 1, hint: what is the derivative oflnk n?arrow_forwardWithout using truth tables, show that ¬p (q r) and q (p r) are logically equivalent. Prove that the sum of two rational numbers is rational using direct proofs. Prove that if n is a positive integer, then n is odd if f is odd.arrow_forwardQuestion 3 Please solve the recurrence and show its proof by induction of: T(1) = 3 T(n) = T(n/3) + 2n, n > 1arrow_forward
- Suppose we have positive integers a, b, and c, such that that a and b are not relatively prime, but c is relatively prime to both a and b . Let n = s × a + t × b be some linear combination of a and b, where s and t are integers. Prove that n cannot be a divisor of c. Follow the definition of relative primes, and use contradiction.arrow_forwardFind the value of x for the following sets of congruence using the Chinese remainder theorem.a. x ≡ 2 mod 7, and x ≡ 3 mod 9b. x ≡ 4 mod 5, and x ≡ 10 mod 11c. x ≡ 7 mod 13, and x ≡ 11 mod 12arrow_forwardUse Direct Proof to prove this.. If x ∈ ℝ and x 2 + x + 6 < 0, then -x 2 + 2x - 9 < 0.arrow_forward
- Informal Proofs Use strong induction to show that every positive integer, n, can be written as a sum of distinct powers of two: 20, 21, 22, 23, ...:1 = 20, 2 = 21, 3 = 20 + 21, ....arrow_forward– You must justify your answers. – The answers should be concise, clear and neat. – When presenting proofs, every step should be justified --- also use topological sortingarrow_forwardInvestigate the role of reducibility in the context of NP completeness and Analyse the following with real world programming examples: Polynomial complexity Exponential complexityarrow_forward
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