img+1-617-874-1011 (US)
img+44-117-230-1145 (UK)
ws+61-7-5641-0117 (AU)
Live chat

Natural Number and Induction Homework Help

1. Show all steps in calculating that 2.(1+3)=8 including justification of each step. Do not use any laws of arithmetic. Only use the definitions of addition and multiplication, together with abbreviations.

Natural Number And Induction img1

2. Prove that for all natural numbers n,


(8 points)

3. Prove by induction on k that for every natural number k, there are natural numbers m and n, so that


(10 points)

4. The Fibonacci function is defined by the following recursion.

Natural Number And Induction img2
  1. Construct a table of the first 10 values of fib(n).
  2. Show that for every natural number n,
  3. fib(2n+2)+fib(n)2=fib(n+2)2



In these problem, we consider lists consisting only of natural numbers.

(10 points)


  1. We want to define the sum of such a list to mean the obvious thing: Σ([2,3,4])=9. Write a formal recursive definition of Σ(K).
  2. Using your definition, prove that for any two lists of natural numbers, Σ(K+L)=Σ(K)+Σ(L).

Sets and Functions

Note: A function f: {'X->Y'} is onto if it is the case that for every b∈Y there is at least one a∈X so that f(a)=b. A function is one-to-one if it is that case that for every b∈Y, there is at most one a∈X so that f(a)=b.

6. Consider the sets {'A={0,1,2,3}'}and {'B={a,b,c}'}.

  1. Write out the set B X A using correct notation.
  2. Draw a picture of a function from A to B that is not a bijection (not a matching).
  3. Is there a bijection between A and B. Explain your answer.
  4. How many functions are there from B to A? Explain your answer.
  5. How many onto functions are there from B to A.

(8 points)


  1. Given two sets C and D for which C X D is equal to D X C.
  2. Show that for any two sets A and B, it is the case that there is a bijection from A X B to B X A.

(6 points)

8. Suppose you are given following functions: {'f:N->N,g:N->N and N->N'} defined by




  1. Using composition only (that is, by writing expressions like g o g o h), define new functions that calculate the following
  2. {'m->m'}8+4




  3. Is it possible to define {'m ->m'}3 using composition and these three functions ? Explain why or why not.

(8 points)

9. We write A ∼ B to mean that A and B are the same size (have the same cardinality). Write a clear and rigorous definition of what A ∼ B means in terms of functions. Do not give examples. Just write a mathematical definition

(6 points)


  1. For sets A and{' B = {0,1}'}, define a one-to-one function {'F:A->B'}A.
  2. Show that F is not also onto.
  3. Show that three is no onto function from A to BA.

(12 points)

11. We say that a set is countable if there is a one-to-one function {'f:A->N'}. We say that a set is denumerable if there is an onto function {'g:N->A'}.

  1. Show that every denumerable set is countable.
  2. Show that every non-empty countable set is denumerable.

(10 points)

24 x 7 Availability.
Trained and Certified Experts.
Deadline Guaranteed.
Plagiarism Free.
Privacy Guaranteed.
Free download.
Online help for all project.

Urgenthomework helped me with finance homework problems and taught math portion of my course as well. Initially, I used a tutor that taught me math course I felt that as if I was not getting the help I needed. With the help of Urgenthomework, I got precisely where I was weak:


  Urgent HomeWork

Disclaimer: The study tools and academic assistance/guidance through online tutoring sessions provided by Urgenthomework.com is to help and enable students to compete academically. The website does not provide ghostwriting services and has ZERO TOLERANCE towards misuse of the services. In case any user is found misusing our services, the user's account will be immediately terminated.
Copyright © 2009-2023 UrgentHomework.com, All right reserved.