Is the following a theorem?
Every infinite recursive subset of is the range of a strictly increasing primitive recursive function.
Is the following a theorem? Every infinite recursive subset of is the range of a strictly...
Show that every infinite Turing-recognizable language is the range of some one-to-one computable function.
Nearest point theorem: letF be a non-void closed subset of Rp and let x be a point outside of F. Then there exists at least one point y belonging to F such that ||z - x|| greater than or equal to ||y -x|| for all z in F. Given the theorem, answer the question: Does the nearest point theorem in R imply that there is a strictly positive real number nearest zero?
Prove that every subset of N is either finite or countable. (Hint: use the ordering of N.) Conclude from this that there is no infinite set with cardinality less than that of N.
Real Analysis II Please do it without using Heine-Borel's theorem and do it only if you're sure Problem: Let E be a closed bounded subset of En and r be any function mapping E to (0,∞). Then there exists finitely many points yi ∈ E, i = 1,...,N such that Here Br(yi)(yi) is the open ball (neighborhood) of radius r(yi) centered at yi. Also, following definitions & theorems should help that E CUBy Definition. A subset S of a topological...
Consider the integer knapsack problem. Give a recursive algorithm (call it Find-Optimal-Subset) that finds the optimal subset of items through post-processing, that is, after filling in the memorization table to find the maximum total value of the optimal subset of items. (The algorithm we studied in class finds only the maximum total value, not the actual optimal subset of items.) Hint: Trace back through the array M[0..n,0..W] following the optimal structure.
A function f:R HR is said to be strictly increasing if f(x1) < f(12) whenever I] < 12. Prove: If a differentiable function f is strictly increasing, then f'(x) > 0. Then give counterexamples to show that the following statements are false, in general. (i) If a differentiable function f is strictly increasing, then f'(2) >0 for all 1. (ii) If f'(x) > 0 for all x, then f is strictly increasing -
answer question 5 please 3 and 4 are just included to refer to the theorems 3 Prove the following theorem: Theorem 2.2. Let S be a ser. The following statements are equivalent: (1) S is a countable set, i. e. there exists an injective function :S (2) Either S is the empty ser 6 or there exists a surjective function g: N (3) Either S is a finite set or there exists a bijective function h: N S (4) Prove...
Hint: Use the fundamental theorem of arithmetic. 15. Theorem 14.5 implies that Nx N is countably infinite. Construct an alternate proof of this fact by showing that the function ф : N x N 2n-1(2m-1) is bijective. N defined as ф(m,n) It is also true that the Cartesian product of two countably infinite sets is itself countably infinite, as our next theorem states. Theorem 14.5 If A and B are both countably infinite, then so is A x B. Proof....
input --> NxN matrix of integers where every row and column strictly increasing. How can I design a search algorithm to determine whether matrix contains a element x? Can you also analyze it? PLEASE make efficient as possible.
2 er Let I be an interval of R, and define the function f :I→ R by f(x) 1 +e2z or every z EZ. (a) Find the largest interval T where f is strictly increasing. (b) For this interval Z, determine the range f(T) (c) Let T- f(I). Show that the function f : I -» T is injective and surjective. (d) Determine the inverse function f-i : T → 1. (e) Verify that (fo f-1)()-y for every y E...