# Theorem: In Any Finite Set Of Women If One Has Blue Eyes Then They All Have Blue Eyes

Theorem: In any finite set of women, if one has blue eyes then they

all have blue eyes.

Proof. Induction on the number of elements. if n= or n=1 it is immediate. Assume it is true for k Consider a group with k+1 women, and without loss of generality assume the first one has blue eyes. I will represent one with blue eyes with a '*' and one with unknown eye color as @. You have the set of women: {*,@,...,@} with k+1 elements. Consider the subset made up of the first k. This subset is a set of k women, of which one has blue eyes. By the induction hypothesis, all of them have blue eyes. We have then: {*,...,*,@}, with k+1 elements. Now consider the subset of the last k women. This is a set of k women, of which one has blue eyes (the next-to-last element of the set), hence they all have blue eyes, in particular the k+1-th woman has blue eyes. Hence all k+1 women have blue eyes. By induction, it follows that in any finite set of women, if one has blue eyes they all have blue eyes. QED

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