Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
expand_more
expand_more
format_list_bulleted
Concept explainers
Question
Chapter 4.6, Problem 3E
Program Plan Intro
To show that the case 3 of the master theorem is overstated.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
Match the following sentence to the best suitable answer:
- A. B. C. D.
for the linear congruence ax=1(mod m), x is the inverse of a, if__________
- A. B. C. D.
What is -4 mod 9 ?
- A. B. C. D.
The solution exists for a congruence ax=b(mod m) such that GCD(a,m)=1 and
- A. B. C. D.
(107+22)mod 10 is equivalent to :_________
A.
5
B.
c divides b
C.
GCD(a,m)=1
D.
9 mod 10
Demonstrate how the following instance of the 3SAT problem can be reduced to its corresponding Sum of Subsets problem. Consequently prove that the Sum of Subsets problem belongs to the class of NPC problems, if 3SAT is in NPC also.
= (xyz) (-x-y-z) (xyz) (-x-yz)
where x=0, y=0, 2-1, and represents negation
) Show that ∀xP(x) ∧ ∃xQ(x) is logically equivalent to ∀x∃y(P(x) ∧ P(y))
The quantifiers have the same non empty domain.
I know that to prove a proposition is logically equivalent to another one, I have to show that
∀xP(x) ∧ ∃xQ(x) ↔ ∀x∃y(P(x) ∧ P(y))
Which means I have to prove that
(∀xP(x) ∧ ∃xQ(x)) → ∀x∃y(P(x) ∧ P(y)) ∧ ∀x∃y(P(x) ∧ P(y)) → (∀xP(x) ∧ ∃xQ(x))
I don't know the answer, so I saw the textbook answer. It says
(1) Suppose that ∀xP(x) ∧ ∃xQ(x) is true. Then P(x) is true for all x and there is an element y for which Q(y) is true.
I get this part.
Because P(x) ∧ Q(x) is true for all x and there is a y for which Q(y) is true, ∀x∃y(P(x) ∧ P(y)) is true.
Emm... I think ∀x∃y(P(x) ∧ P(y)) is true because ∀x only affects P(x) and ∃y only affects P(y) since their alphabets are different. So, it has the exact same meaning as ∀xP(x) ∧ ∃yQ(y). And since the domains are the same, ∀xP(x) ∧ ∃yQ(y) is actually equal to ∀xP(x) ∧ ∃xQ(x).
But the textbook states that "P(x) ∧ Q(x) is…
Chapter 4 Solutions
Introduction to Algorithms
Ch. 4.1 - Prob. 1ECh. 4.1 - Prob. 2ECh. 4.1 - Prob. 3ECh. 4.1 - Prob. 4ECh. 4.1 - Prob. 5ECh. 4.2 - Prob. 1ECh. 4.2 - Prob. 2ECh. 4.2 - Prob. 3ECh. 4.2 - Prob. 4ECh. 4.2 - Prob. 5E
Ch. 4.2 - Prob. 6ECh. 4.2 - Prob. 7ECh. 4.3 - Prob. 1ECh. 4.3 - Prob. 2ECh. 4.3 - Prob. 3ECh. 4.3 - Prob. 4ECh. 4.3 - Prob. 5ECh. 4.3 - Prob. 6ECh. 4.3 - Prob. 7ECh. 4.3 - Prob. 8ECh. 4.3 - Prob. 9ECh. 4.4 - Prob. 1ECh. 4.4 - Prob. 2ECh. 4.4 - Prob. 3ECh. 4.4 - Prob. 4ECh. 4.4 - Prob. 5ECh. 4.4 - Prob. 6ECh. 4.4 - Prob. 7ECh. 4.4 - Prob. 8ECh. 4.4 - Prob. 9ECh. 4.5 - Prob. 1ECh. 4.5 - Prob. 2ECh. 4.5 - Prob. 3ECh. 4.5 - Prob. 4ECh. 4.5 - Prob. 5ECh. 4.6 - Prob. 1ECh. 4.6 - Prob. 2ECh. 4.6 - Prob. 3ECh. 4 - Prob. 1PCh. 4 - Prob. 2PCh. 4 - Prob. 3PCh. 4 - Prob. 4PCh. 4 - Prob. 5PCh. 4 - Prob. 6P
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.Similar questions
- Justify that the Master theorem may be used for solving recurrences of the specified form. Solve thefollowing recurrence relation using the Master theorem:T (n) = 9 T(n/ 3) + narrow_forwardLet R=ABCDEGHK and F= {ABK→C, A→DG, B→K, K→ADH, H→GE} . Is it in BCNF? Prove your answer.arrow_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_forward
- Show that ¬ (p ∨ (¬p ∧ q)) and ¬p ∧ ¬q are logically equivalent by developing a series of logical equivalences.arrow_forwardSolve this proof with no premises: (~Q->~P)->((~Q->P)->Q)arrow_forwardprove the equivalence of the schemata:~(~(p • ~ q ) • ~ ( ~ p• q)) equal to ~(p • q) • ~(~p • ~q)arrow_forward
- Write a Hilbert proof for the following theorem schema ⊢((p→q)∨(p∧(¬q)))arrow_forward3) Explain in detail about P, NP, NP-complete and NP-hard problems with real time examplesforeach? 4) Show that the Post Correspondence Problem is decidable over unary alphabetarrow_forwardPlease answer the following question in depth with full detail. Suppose that we are given an admissible heuristic function h. Consider the following function: 1-h'(n) = h(n) if n is the initial state s. 2-h'(n) = max{h(n),h'(n')−c(n',n)} where n' is the predecessor node of n. where c(n',n) min_a c(n',a,n). Prove that h' is consistent.arrow_forward
- Simplify the given proposition using Logical Equivalence rules and determine if it is a tautology, contradiction orcontingency. Do not forget to indicate rule/s applied in each step. (q → ¬p) ↔ parrow_forwardSuppose you want to solve the following equality 2a + b + 3c + 4d + 6e = 45 What is the chromosome phenotype? What is the fitness function? What is the fitness value of a, b, c, d, e = (The first five numbers of university ID)? (hint if ID= 437818854 then a=1, b=8, c=8, d=5, e=4)arrow_forwardUsing the Master Theorem, find the order of growth of the following recurrence relations.(i) M(n) = 6M(n/4) + 9n3 + 4n, M(1) = 2(ii) M(n) = 3M(n/5) + 2n3 – 3n log n , M(1) = 1(iii)M(n) = 2M(n/2), M(1) = 0arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Operations Research : Applications and AlgorithmsComputer ScienceISBN:9780534380588Author:Wayne L. WinstonPublisher:Brooks Cole
Operations Research : Applications and Algorithms
Computer Science
ISBN:9780534380588
Author:Wayne L. Winston
Publisher:Brooks Cole