What's wrong with this proof for the set C={a1}?

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Homework Statement



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Homework Equations





The Attempt at a Solution



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Why can't C={a1}?
 
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Which part of
The reason is that C = {a1} = B, so an element of A, namely a2, is not in either B or C.
don't you understand?
 
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The proof clearly falls apart when it is stated let Set ##A## be a set of ##k+1## numbers. These number can either be all the same, or different. Hence the proof is invalid from that point on.
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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