×
Log in to StudySoup

Forgot password? Reset password here

Use mathematical induction (and the proof of Proposition

Discrete Mathematics with Applications | 4th Edition | ISBN: 9780495391326 | Authors: Susanna S. Epp ISBN: 9780495391326 48

Solution for problem 1E Chapter 5.2

Discrete Mathematics with Applications | 4th Edition

  • Textbook Solutions
  • 2901 Step-by-step solutions solved by professors and subject experts
  • Get 24/7 help from StudySoup virtual teaching assistants
Discrete Mathematics with Applications | 4th Edition | ISBN: 9780495391326 | Authors: Susanna S. Epp

Discrete Mathematics with Applications | 4th Edition

4 5 0 380 Reviews
12
5
Problem 1E

Use mathematical induction (and the proof of Proposition as a model) to show that any amount of money of at least 14c can be made up using 3¢ and 8¢ coins.

Proposition

For all integers n ≥ 8, n¢ can be obtained using 3¢ and ¢ coins.

Proof (by mathematical induction):

Let the property P(n) be the sentence

n¢ can be obtained using 3¢ and 5¢coins. ← P(n)

Show that P(8) is true:

P(8) is true because 8¢can be obtained using one 3¢coin and one 5¢ coin.

Show that for all integers k≥ 8, if P(k) is true then P(k+1) is also true:

[Suppose that P(k) is true for a particular but arbitrarily chosen integer k ≥ 8. That is:]

Suppose that k is any integer with k ≥ 8 such that

k¢ can be obtained using 3¢ and 5¢ coins. ← P(k) inductive hypothesis

[We must show that P(k + 1) is true. That is:] We must show that

(k + 1)¢can be obtained using 3¢ and 5¢ coins. ← P(k + 1)

Case 1 (There is a 5¢ coin among those used to make up the k¢.): In this case replace the 5¢ coin by two 3¢ coins; the result will be (k + 1) ¢.

Case 2 (There is not a 5¢ coin among those used to make up the k¢.): In this case, because k ≥ 8, at least three 3¢ coins must have been used. So remove three 3¢ coins and replace them by two 5¢ coins; the result will be (k + 1) ¢.

Thus in either case (k + 1) ¢ can be obtained using 3¢ and 5¢ coins [as was to be shown].

[Since we have proved the basis step and the inductive step, we conclude that the proposition is true.]

Step-by-Step Solution:
Step 1 of 3

ISS 225 CLASS NOTES VOCABULARY Abrogate - to abolish or do away with Court of Appeals - sit on panel of 3 Court Order - allows one party of a case to carry certain steps, this does not require probable cause Dicta - reasoning from a case that does not affect the holding, usually acknowledging a hypothetical situation that the court will not and/or does not need to respond to, if responded to it...

Step 2 of 3

Chapter 5.2, Problem 1E is Solved
Step 3 of 3

Textbook: Discrete Mathematics with Applications
Edition: 4
Author: Susanna S. Epp
ISBN: 9780495391326

Unlock Textbook Solution

Enter your email below to unlock your verified solution to:

Use mathematical induction (and the proof of Proposition

×
Log in to StudySoup
Get Full Access to Math - Textbook Survival Guide

Forgot password? Reset password here

Join StudySoup for FREE
Get Full Access to Math - Textbook Survival Guide
Join with Email
Already have an account? Login here
Reset your password

I don't want to reset my password

Need an Account? Is not associated with an account
Sign up
We're here to help

Having trouble accessing your account? Let us help you, contact support at +1(510) 944-1054 or support@studysoup.com

Got it, thanks!
Password Reset Request Sent An email has been sent to the email address associated to your account. Follow the link in the email to reset your password. If you're having trouble finding our email please check your spam folder
Got it, thanks!
Already have an Account? Is already in use
Log in
Incorrect Password The password used to log in with this account is incorrect
Try Again

Forgot password? Reset it here