8. Explain why the following "proof" is incorrect.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Help me please
Transcribed Image Text:8. Explain why the following "proof" is incorrect.
Proposition 0.1. All dogs are the same color.
Proof. We will prove this proposition using mathematical induction. For each natural
number
n, we set
P(n): Any set of n dogs consists entirely of dogs of the same color.
We will prove that for each natural number n, P(n) is true, which will prove that all
dogs are the same color.
Base case: A set with only one dog consists entirely of dogs of the same color and,
hence, P(1) is true.
Inductive step: Let k be a natural number and assume that P(k) is true, that is,
that every set of k dogs consists of dogs of the same color. Now consider a set D of
k+1 dogs, where
D = {d1, d2,. .
dk+1}.
%3D
If we remove the dog di from the set D, we then have a set D1 of k dogs, and using the
assumption that P(k) is true, these dogs must all be of the same color. Similarly, if we
remove de+1 from the D, we again have a set D2 of k dogs, and these dogs must all be
the same color. Furthermore, because dogs d2, . ., dk are in both sets D1 and D2, the
dogs in D1 and D2 must be the same color. Since D = D1 U D2, we have proved that
all of the dogs in D must be of the same color. This proves that if P(k) is true, then
P(k +1) is true and, hence, by mathematical induction, we have proved that for each
natural number n, any set of n dogs consists entirely of dogs of the same color.
....
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
