Question

What is wrong with the following "proof" that all roses are the same color. It suffices to prove the statement: In every set of n roses, all the roses in

the set are the same color. If n = 1, the statement is certainly true. Assume

the statement is true for n = k. Let S be a set of k + 1 roses. Remove one

rose (call it rose A) from S; there are k roses remaining, and they must all

be the same color by the induction hypothesis. Replace rose A and remove

a different rose (call it rose B). Once again there are k roses remaining that

must all be the same color by the induction hypothesis. Since the remaining

roses include rose A, all the roses in Shave the same color. This proves that

the statement is true when n = k + 1. Therefore, the statement is true for all

n by induction.

Short answer

Alln + 1 roses in $\left\{R_2,R_3,....R_{n+1}\right\}$ are the same color.

Step-by-step solution

Step 1

Consider the following statement P (n) as following;

In every set of roses of size n, all n roses are the same color.

Step 2

Assume that, P (k) for some $k\ge 1$. Now, $\left\{R_2,R_3,....R_n\right\}$ is a set of n roses, the induction hypothesis applies to this set. Thus, all the roses in this set are the same color.

Step 3

Since, $\left\{R_2,R_3,....R_{n+1}\right\}$is also a set of n roses, the induction step likewise holds for set.

Thus, all the roses in this set are the same color too.

Therefore, all n+ 1 roses in $\left\{R_2,R_3,....R_{n+1}\right\}$are the same color.