# Mathematical Induction Question

Question1:
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)
It is a statement of the form nℕ, P(n)
What is the predicate function P(n)?

b)    Proof: Base cases (4 marks)

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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_2⁡n
a) Predicate function (1 mark)
It is a statement of the form nℕ+, 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).

Pages (550 words)
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