# Advanced Math Problems

## Transcribed Text

1. Prove that for any a, b ∈ C, |a − b| 2 + |a + b| 2 = 2(|a| 2 + |b| 2 ). What is geometric interpretation of this equality ? 2. Given two points z1 and z2 in C, draw the picture of the set A = {z | |z − z1| = |z − z2|}. 3. Is the set A = {z = x + iy | − 1 < x < 1, −1 < y < 1} an open set? Verify your conclusion. 4. Let A = {aj} ∞ j=1 be a sequence of point in the unit disk ∆(1) such that aj → a ∈ ∆(1). Whether or not ∆(1) − A is an open set? If yes, tell the reason; if not, give a counterexample. 5. Let A = { 1 n | n = 1, 2, 3, ...}. Is A an open set ? Is A compact or a closed set? You need to tell the reason.

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