- #1
jr16
- 14
- 0
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.
Show that Un [itex]\subseteq[/itex] U2n for every positive integer, n.
[1] Un = {z ε ℂ, zn = 1}
[2] Un = {cos([itex]\frac{2m\pi}{n}[/itex]) + i sin([itex]\frac{2m\pi}{n}[/itex])}
First I started out by comparing the two sets using the first equation:
(i)zn = 1
(ii)z2n = 1
(zn)2 = 1
zn = [itex]\sqrt{1}[/itex]
zn = [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!
Homework Statement
Show that Un [itex]\subseteq[/itex] U2n for every positive integer, n.
Homework Equations
[1] Un = {z ε ℂ, zn = 1}
[2] Un = {cos([itex]\frac{2m\pi}{n}[/itex]) + i sin([itex]\frac{2m\pi}{n}[/itex])}
The Attempt at a Solution
First I started out by comparing the two sets using the first equation:
(i)zn = 1
(ii)z2n = 1
(zn)2 = 1
zn = [itex]\sqrt{1}[/itex]
zn = [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!