Can the Horse Induction Proof Truly Confirm All Horses Are the Same Color?

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Homework Statement

Let P(n) be the statement "In every set of n horses, all of the horses in the set have the same colour."
Base Case: We must prove that P(1) is true. If our set only contains one horse, then all horses in the set have the same colour.
Inductive Step: Let m ≥ 1 and assume P(m) is true. For any set of m horses, all m horses in the set have same colour. We will prove that P(m+1) is true. Let S be a set of m+1 horses named

x₁, x₂,..., x_m+1.

The horses

x₁, x₂,..., x_m

are a set of m horses. By the inductive hypothesis, all of the horses in this set have the same colour.

Furthermore, the horses

x₁, x₃,..., x_m+1

are a set of m horses. By the Inductive Hypothesis, all of the horses

x₁, x₃,..., x_m+1

in this set have the same colour.

It follows that all m+1 horses

x₁, x₂,..., x_m+1

are the same colour, because they are all the same colour as horse x₁.

Therefore, P(m+1) is true.

By induction, P(n) is true for all positive integers n. That is, all horses have the same colour.

What is wrong with this proof?

Homework Equations

The Attempt at a Solution

I am very confused about why there is no longer an x₂ in the one part of the question where it goes

"Furthermore, the horses

x₁, x₃,..., x_m+1

are a set of m horses. By the Inductive Hypothesis, all of the horses

x₁, x₃,..., x_m+1

in this set have the same colour."

Why is that?

And is the proof wrong because it would not work if there is two horses? Or when m is equal to 1?

 

[PeroK]

And is the proof wrong because it would not work if there is two horses? Or when m is equal to 1?

Yes, the inductive step assumes at least 2-3 horses.