Question 1:

In this question you will use strong induction to prove that your new algorithm works correctly.

In other words, you will prove that for all n element of ℕ for all x element of ℝ-{0} FP(x,n) = xn

a) Predicate Function (1 mark)

Your conjecture has already been stated in symbolic form:

It is a statement of the form nℕ, P(n)

What is the predicate function P(n)?

b) Proof: Base cases (4 marks)

Don't use plagiarized sources. Get Your Custom Essay on

Mathematical Induction Question

From as low as $9/Page

c) Proof: Inductive step setup (2 marks)

This is the beginning of the inductive step where you are stating the assumptions in the inductive step and what you will be proving in that step. As you do so, identify the inductive hypothesis.

d) Proof: Inductive step (14 marks)

Question 2:

In this question you will use strong induction to prove that your new algorithm is very efficient.

Given a non-zero real number x, and a natural number n, define CFP(x,n) to be the cost of FP(x,n) = the total number of multiplications in the total execution of FP(x,n)

You will prove that for all n element of ℕ+ all x element of ℝ-{0} CFP(x,n) <= 2 log_2n

a) Predicate function (1 mark)

Your conjecture has already been stated in symbolic form:

It is a statement of the form nℕ+, P(n)

What is the predicate function P(n)?

b) Proof: Base cases (2 marks)

Proof: Inductive step setup (2 marks)

This is the beginning of the inductive step where you are stating the assumptions in the inductive step and what you will be proving in that step. As you do so, identify the inductive hypothesis.

d) Proof: Inductive step (17 marks)

Question 3: Define a game as follow: you begin with an urn that contains a mixture of black and white balls, and during the game you have access to as many extra black and white balls as you need.

In each move of the game, you remove two balls from the urn without being able to see what colour they are. Then you look at their colour and do the following:

If the balls are the same colour, you keep them out of the urn and put a black ball in the urn.

if the balls are different colours, you keep the black one out of the urn and put the white one back into the urn.

Each move reduces the number of balls by one, and the game will end when only one ball is left in the urn.

In this assignment you will figure out how to predict the colour of the last ball in the urn and prove your answer using mathematical induction.

a) Make a conjecture about the colour of the final ball based on the initial number of black and white balls in the urn.

b) Translate that conjecture into a theorem in symbolic form using first order logic notation. You will need to invent some notation, including functions, to do so. Define your new notation and functions clearly.

c) Use mathematical induction to prove the formal conjecture you made in Q2.

Before you start, please identify the predicate function P(n) that you will be proving

In the inductive step of your proof, do not forget to clearly identify the Inductive Hypothesis (IH). For more information on Mathematical Induction Question check out : https://www.britannica.com/science/mathematical-induction

Our Advantages

Plagiarism Free Papers

Thehomeworkwritings.com’s team of writers write all papers from scratch. We deliver 100% original, unique papers. That’s what makes us the best custom homework writing service

Free Revisions

We provide unlimited free revisions to all customers and on all papers. Try The Homework Writings today for the best custom homework writing service and experience in the industry.

Title-page

Thehomeworkwritings.com gives clients title pages free of charge. Your only job is to fill out our order form. We will handle the rest.

Bibliography

As the leading essay writing service, we never submit any paper without a reference/bibliography page. We do this free of charge too.

Originality & Security

At Thehomeworkwritings.com, we take great pride in delivering only high-quality 100% original papers to all our clients. We also never share any of our clients’ information with third parties. Your data is safe with us.

24/7 Customer Support

No other custom homework writing service has a friendly, always available customer support team to respond to clients like us.

Try it now!

How it works?

Follow these simple steps to get your paper done

Place your order

Fill in the order form and provide all details of your assignment.

Proceed with the payment

Choose the payment system that suits you most.

Receive the final file

Once your paper is ready, we will email it to you.

Why outsource our services

We have the best customer support team for your essay writing needs.

Pricing

You won’t find any other custom homework writing service with pricing as flexible and affordable as ours.

Communication

Admission help & Client-Writer Contact

We provide a direct line of communication with our writers for the best writing experience.

Deadlines

Paper Submission

As the leading custom homework writing service, we take deadlines very seriously. You will have your paper submitted on time without any delays.

Reviews

Customer Feedback

We truly value your feedback, good or bad, and always use your feedback to help us provide you with an even better custom homework writing service