Hi there,
I have these complex unit Cicle questions.
Let O = {w \mathbb{C}||z|=1}
Let the sets alpha, beta, delta and gamma be defined as the following.
alpha = {w \mathbb{C}| \exists z_1 \in O \exists z_2 \in O : w = z_1 + z_2}
beta = {w \mathbb{C}| \forall z_1 \in O \forall z_2 \in O : w = z_1 + z_2}
delta = {w \mathbb{C}| \exists z_1 \in O \forall z_2 \in O : w = z_1 + z_2}
gamma = {w \mathbb{C}| \forall z_1 \in O \exists z_2 \in O : w = z_1 + z_2}
Solution(alpha)
If e^{i \theta} + e^{-i \theta} for 0 \leq \theta \leq \pi/2 belongs to alpha
hence [0,2] \subseteq alpha
if z \in [0,2] then e^{i \theta}z \in alpha
for all 0 \leq \theta \leq 2 \pi
hence {z: |z| \leq 2} \subseteq alpha.
Therefore no complex larger than two belongs to alpha.
No I am in shady waters here with the remaining set. i
beta solution
Isn't this the same as alpha?
delta solution
if z_1 belongs to [0,1], and if z_2 belong to [0,2], then e^{i \theta} z_1 belongs to delta and e^{i \theta} z_2 belongs to delta.
Then as long as {w:|z_1 + z_2| \leq 2}
No complex number larger than 2 belongs to delta.
How do this sound?
Best Regards
Billy