Hey everyone! I would really appreciate some help with this problem. I have been racking my brain for hours now, and nothing seems to work/convince me.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

Show that U_{n}[itex]\subseteq[/itex] U_{2n}for every positive integer, n.

2. Relevant equations

[1] U_{n}= {z ε ℂ, z^{n}= 1}

[2] U_{n}= {cos([itex]\frac{2m\pi}{n}[/itex]) + i sin([itex]\frac{2m\pi}{n}[/itex])}

3. The attempt at a solution

First I started out by comparing the two sets using the first equation:

(i)z^{n}= 1

(ii)z^{2n}= 1

(z^{n})^{2}= 1

z^{n}= [itex]\sqrt{1}[/itex]

z^{n}= [itex]\pm[/itex]1

But I was not sure if that was enough to show one is a subset of the other

So, then I tried using the second formula

(i) [itex]\Theta[/itex]_{n}= [itex]\frac{2m\pi}{n}[/itex]

(ii) [itex]\Theta[/itex]_{2n}= [itex]\frac{m\pi}{n}[/itex]

I hoped I could somehow deduce that given the above theta values, one must be a subset of the other

But unfortunately, I am not sure if I am going about this proof in the right manner. I would really love any guidance you could give me. Thank you in advance!

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# Homework Help: Roots of Unity Proof

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