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

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.

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%26lt; or = IwI
z~2 w if IzI%26lt; or = IwI


1. if z ~ w then |z|%26lt;= |w|, but we cannot reverse the order,


i.e. |w| might not be %26lt;= |z|, therefore w ~ z might not be true. example:


z=1, w = 1+i,


then z~ w since |z| = 1%26lt; sqrt(2) = |w|


but w is not~z, since |w| is not %26lt;= than 1.


2. z~z is satisfied


3. z~w, and w~y, then |z|%26lt;=|w| and |w| %26lt;= |y| =%26gt; |z| %26lt;|y| =%26gt; z~y.
Reply:The first one I will prove one of the three properties.





if z1 ~1 z2, then 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