Theorem. All horses are the same color.Proof. We'll induct on the number of horses. Base case: 1 horse. Clearly withjust 1 horse, all horses have the same color.Now, for the inductive step: we'll show that if it is true for any group of Nhorses, that all have the same color, then it is true for any group of N+1 horses.Well, given any set of N+1 horses, if you exclude the last horse, you get a set ofN horses. By the inductive step these N horses all have the same color. But byexcluding the first horse in the pack of N+1 horses, you can conclude that thelast N horses also have the same color. Therefore all N+1 horses have the samecolor. QED.Hmmn... clearly not all horses have the same color. So what's wrong with thisproof by induction?

The base case in the induction is wrong. The base case is with N =1 horse. Therefore if we take…

Theorem. All horses are the same color. Proof. We'll induct on the number of horses. Base case: 1 horse. Clearly with just 1 horse, all horses have the same color. Now, for the inductive step: we'll show that if it is true for any group of N horses, that all have the same color, then it is true for any group of N+1 horses Well, given any set of N+1 horses, if you exclude the last horse, you get a set o N horses. By the inductive step these N horses all have the same color. But by excluding the first horse in the pack of N+1 horses, you can conclude that the last N horses also have the same color. Therefore all N+1 horses have the same color. QED. Hmmn... clearly not all horses have the same color. So what's wrong with this proof by induction?

Theorem. All horses are the same color. Proof. We'll induct on the number of horses. Base case: 1 horse. Clearly with just 1 horse, all horses have the same color. Now, for the inductive step: we'll show that if it is true for any group of N horses, that all have the same color, then it is true for any group of N+1 horses Well, given any set of N+1 horses, if you exclude the last horse, you get a set o N horses. By the inductive step these N horses all have the same color. But by excluding the first horse in the pack of N+1 horses, you can conclude that the last N horses also have the same color. Therefore all N+1 horses have the same color. QED. Hmmn... clearly not all horses have the same color. So what's wrong with this proof by induction?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

telephone numbers in particular area consist of a three-digit area code followed by a seven- digit local number. Suppose neither the ﬁrst digit of an area code nor the ﬁrst digit of a local number can be a zero but that all other choices are acceptable. How many different area codes are possible? For a given area code, how many local telephone numbers are possible? How many telephone numbers are possible?
Discrete Mathematics:In how many ways can 1 senator and 3 legislative members be chosen from a group of 20 people. (Lets assume the senator isn't a member in the legislative.)
10 identical copies of a movie will be stored on 40 computers such that each computer has at most 1 copy. How many different ways can the 10 copies be stored? O P(40, 10) O 1040 O 4010
Suppose you have infinitely many cards; and on each card a number is written from 1 to 5. (there are 5 types of cards) How many ways can you choose 10 cards from that deck? (order not important)Ex: You choose three 1s, three 2s, four 5s OR You choose ten 3s O a. 5! x 105 O b. 105 O c. C(10,5) / P(5,5) O d. C(14,4) O e. 510 O f. P(14,10) O g. C(10,5) O h. C(15,5)
In a group of 11 students, there are 6 females and 5 males How many different 4-member committees are possible if there are no restrictions? How many different 4-member committees are possible there must be exactly 2 girls? How many different 4-member committees are possible there must be at least 1 girl? Submit Question
18
A restaurant offers a special pizza with any 4 toppings. If the restaurant has 16 topping from which to choose, how many different special pizzas are possible?special pizzas
I have a permutations & combinations geometry question.
N5 Suppose that you have a group of 10 people (where any two people are either friends or enemies) and one person has at least 6 enemies. Prove that there are either three mutual friends or four mutual enemies.
Q11: PERMUTATIONS