Question: Use mathematical induction to prove the following statement: Consider an alphabet S = {a,b}. Recall that S^n is the set of all strings over S

Use mathematical induction to prove the following statement:

Consider an alphabet S = {a,b}. Recall that S^n is the set of all strings over S of length n. Define S^{<=n} as union of S^i for i<=n. Then S^{<=n} has (2^{n+1} - 1) elements.

1. Identify P(n), the property on natural numbers to prove the statement.

2.0. Write the base case of induction.

2.1. Prove the base case of induction.

3. Write the induction hypothesis.

4. Prove the induction step.

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock blur-text-image

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Mathematics Questions!

section 1; Paper size Letter 1. Fill in the blanks so that the resulting statement is true. A function f is continuous at a when three conditions are satisfied. . f is defined at so that is a real...

1. Use mathematical induction to prove the following theorem for all integers n >= 2. (15 points) 1 +8+ ... +81 = (8n+1 - 1)/7 2. The result from the problem above can be used to count the number of...

MATH 3210 Practice Final Exam for Discrete I You now have several quizzes and hour exams to study. The following in mainly a repetition of your previous practice and actual exams. 1. (a) Use truth...

For the alphabet A={0,1}, define the set S = {all strings over A formed by concatenating 1 or more copies of 0111 or 1101} Use mathematical induction to prove (base case and inductive case) that for...

1) Is the following argument valid? Zack is either a pianist or a writer. If he is a pianist, then he has a keen sense of hearing. Zack does not have a keen sense of hearing, so he is a writer. The...

Let A, B be sets. Define: (a) the Cartesian product (A B) (b) the set of relations R between A and B (c) the identity relation A on the set A [3 marks] Suppose S, T are relations between A and B, and...

Let r and s be solutions to the quadratic equation x 2 b x + c = 0. For n N, define d0 = 0 d1 = r s dn = b dn1 c dn2 (n 2) Prove that dn = r n s n for all n N. [4 marks] (b) Recall that a commutative...

1: Refinancing a loan. Suppose someone takes out a home improvement loan for $30,000. The annual interest on the loan is 6% and is compounded monthly. The monthly payment is $600. Let an denote the...

Consider Hoare triples of the form {T} V := E {V = E} where T is the atomic formula 'true' and V and E range over variables and expressions, respectively. (i) Write down an instance of such a triple...

I need help with Picture 1 Chapter 2, problems 5, 7v. Chapter 2, problem 7i, use mathematical induction, Picture 2 and 3 Problems 2-7, 10, 11, 15, 18 oces without DOD. 5. Consider the language S*,...

The potentially valid arguments for tariff protection are also the most easily abused. What are those arguments? Why are they susceptible to abuse? Evaluate the use of artificial trade barriers, such...

1) Which of the following statement is false regarding Crohn's Disease? A. It extends throughout the entire GI tract B. The classic symptom is constipation c. There are periods of exacerbation and...

E4.2 A closed-loop system is used to track the sun to obtain maximum power from a photovoltaic array. The tracking system may be represented by a unity feedback control system and 100 Ge(s)G(s) =...

During 2008, Pet Kingdom made estimated tax payments of $56,000 each quarter to the IRS. Determine Pet Kingdom's income tax liability for tax year 2008 providing the following information that would...

A cylindrical proportional tube has an anode wire radius of 0.0045 cm and a cathode radius of 3.3 cm. It is operated with an applied voltage of 2160 V. If a minimum electric field of 1.1 MV/m is...

Find the temperature u(x,t) from the heat equation in a laterally insulated copper bar 80 cm long with C 2 =1.158 cm 2 /sec If the initial temperature is represented by the below graph and the ends...

Philip and Justine both receive taxable social security benefits. In 2019, the amount shown in box 5 of Philip's Form SSA-1099 was $25,078. Justine continues to work and was required to repay...

Question 7 . Given the following simple maze, convert the maze into a graph. ( 5 points ) Question 8 . Implement depth first search in the maze function in homework 1 . py . ( 2 0 points ) Question 9...

2. Take a familiar problem as your starting point. What are the main elements of the problem? Discuss whether you all agree on the choice of main elements.

2. In this chapter, the reader should reflect on the following concepts: Context, winwin, win-lose, ambiguity, mental model, point of reference.

2. In this chapter, the reader should reflect on the following concepts: Patterns, symmetrical relationships, complementary relationships.