2 relations ~1 and ~2 are defined on the complex numbers C by; z~1 w if Re z= Re w and z~2 w if IzI< or = IwI
Question: for each of the relations state whether they are reflexive, symmetric and transitive, and state why they are or are not. so which one would be an equivalence relation.
Answers:
z~2 w if IzI< or = IwI
1. if z ~ w afterwards |z|<= |w|, but we cannot reverse the order,
i.e. |w| might not be <= |z|, in consequence w ~ z might not be true. example:
z=1, w = 1+i,
then z~ w since |z| = 1< sqrt(2) = |w|
but w is not~z, since |w| is not <= than 1.
2. z~z is content
3. z~w, and w~y, then |z|<=|w| and |w| <= |y| => |z| <|y| => z~y.
The first one I will prove one of the three properties.
if z1 ~1 z2, afterwards Re(z1) = Re(z2). This means Re(z1) = Re(z2) so z2 ~ z1.
For the second one, consider this same property and see if it holds for ~2
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