2) Show that the set that contains all the subsets of the natural numbers N (i.e....
Algorithms
8. The problem 'SET COVER gives two numbers n, k, and a family of n subsets of (,.. n It asks whether it is possible to select k of these subsets such that each number in (1,... ,n) occurs in at least one of the selected subsets. (8.1) Show that the problem 'SET COVER' is in the class NP (8.2) The simplest algorithm to solve set cover just tests all the possible choices of k subsets. How long will...
4) Show that the set of all numbers that are not solutions of polynomial equations with integer coefficients is an uncountable set. Hint: Show that if A is uncountable, B is countable and A- BUC then C is uncountable.
Let Rj be the set of all the positive real numbers less than 1, i.e., R1 = {x|0 < x < 1}. Prove that R1 is uncountable.
3. Show that (1.2)+(2-3)+(3.4) + ... + n(n+1) = n(n+1)(n+2) for all natural numbers n = 1,2,3,... 3 4. Show that n2 + 3n is divisible by 2 for all natural numbers n 2 1
Prove that all sets with n elements have 2n subsets. Countthe empty set ∅ and the whole set as subsets.
4. Show that n2 + 3n is divisible by 2 for all natural numbers n 21
b and c please explian thx
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post the question from the book
Let 2 be a non-empty set. Let Fo be the collection of all subsets such that either A or AC is finite. (a) Show that Fo is a field. Define for E e Fo the set function P by ¡f E is finite, 0, if E is finite 1, if Ec is finite. P(h-10, (b) If is countably infinite, show P is finitely additive but not-additive. (c)...
(a) Prove that, for all natural numbers n, 2 + 2 · 2 2 + 3 · 2 3 + ... + n · 2 n = (n − 1)2n+1 + 2. (b) Prove that, for all natural numbers n, 3 + 2 · 3 2 + 3 · 3 3 + ... + n · 3 n = (2n − 1)3n+1 + 3 4 . (c) Prove that, for all natural numbers n, 1 2 + 42 + 72...
13. An algebraic number is a real number which is the root of a polynomial co + ciz c2n in which all of the coefficients c i 1,2,.,n) are integers. The order of an algebraic number is the smallest natural number n for which z is a root of an n-th degree polynomial with integer coefficients. A real number is transcendental if it is not algebraic. a) Show that the set of algebraic numbers of order n is countable (b)...
3. Show that (1-2) + (2.3) + (3.4) + ... + n(n+1) = for all natural numbers n = 1,2,3,... n(n+1)(n+2) 3