Let n ∈ Z^+ and denote by N^n =N×N×...×N (n times). Prove that N^n is countable for all n ∈Z+.
Please answer questions in clear hand-writing and show me the full process, thank you (Sometimes I get the answer which was difficult to read).
Every natural number has a unique representation of product of prime numbers(with power)
p1,p2,... ,pn be the 1st n prime numbers.
Let n ∈ Z^+ and denote by N^n =N×N×...×N (n times). Prove that N^n is countable...
Could you please help me to solve the problem. Also, could you please answer questions in clear hand-writing and show me the full process, thank you (Sometimes I get the answer which was difficult to read).Thanks a lot What is the smallest positive value of n, where n is an integer, such that Algorithm A, whose running time is 100n2 runs faster than Algorithm B, whose running time is 2n , on the same machine (give your answer in whole number(s))
Could you please help me to solve the problem. Also, could you please answer questions in clear hand-writing and show me the full process, thank you (Sometimes I get the answer which was difficult to read).Thanks a lot Suppose we are comparing implementations of two algorithms on the same machine. For input size of n, Algorithm A runs in 8n^2 steps, while Algorithm B runs in 64nlog2(n) steps. For what value n>2, where n is an integer, does Algorithm A...
4. Let n be a positive integer. Z" is the set of all lists of length n whose entries are in Z. Prove that Z" is countable. (Hint: Find a bijection between Z"-1x Z and Z" and then use induction.) 4. Let n be a positive integer. Z" is the set of all lists of length n whose entries are in Z. Prove that Z" is countable. (Hint: Find a bijection between Z"-1x Z and Z" and then use induction.)
*Let . ., A, denote the eigenvalues of an n x n matrix A. Prove that the Frobenius 5. norm of A satisfies ΑIFΣ. i=1 *Let . ., A, denote the eigenvalues of an n x n matrix A. Prove that the Frobenius 5. norm of A satisfies ΑIFΣ. i=1
2. Consider the relation E on Z defined by E n, m) n+ m is even} equivalence relation (a) Prove that E is an (b) Let n E Z. Find [n]. equivalence relation in [N, the equivalence class of 3. We defined a relation on sets A B. Prove that this relation is an (In this view, countable sets the natural numbers under this equivalence relation). exactly those that are are 2. Consider the relation E on Z defined by...
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Define dsup(f,g)-sup |f(t)-g(t) and di(f.g)f (t)- g(t)ldt (f,g E C) tEI (i) Prove that dsup is a metric on C (ii) Prove that di is a metric on C. (You may use any standard properties of continuous functions and integrals, provided you make your reasoning clear.) 6 i) Let 1 denote the constant function on I with value 1. Give an explicit...
(2) For an integer n, let Z/nZ denote the set of equivalence classes [k) tez: k -é is divisible by n (a) Prove that the set Z/nZ has n elements. (b) Find a minimal set of representatives for these n elements. (c) Prove that the operation gives a well-defined addition on Z/nZ Hint: The operution should not depend on the choice of coset representatives Verify that this gives Z/n2 the structure of an ahelian group. Be sure to verify all...
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
Help please! Let {Xn}n=0 be a process taking values in a countable [0, 1]E and stochastic set E, and assume that for some probability vector X matriz P E(0, 1ExE we have prove that Xn ~ Markov(λ, P)
After reading the questions carefully, please prove and compute the questions with clear hand writing. I need to understand clearly, so when you prove these questions, please prove it step by step clearly!!!! 3- Let f: [0,1R be defined by f(x) = x2. For each n e N, let P be the partition of [0, 1 into n equal subintervals 3-1) Find formulas for U (f, P,) and L(f, P,). You may use the formula 2 = " n)without proof....