On the set `C` of all complex numbers an operation `'o'` is defined by `z_1\ o\ z_2=sqrt(z_1z_2)` for all `z_1,\ z_2 in C` . Is `o` a binary operation on `C` ?

Is * defined by a * b=(a+b)/2 is binary operation on Z.

On the set Z of all integers a binary operation ** is defined by a**b=a+b+2 for all a ,\ b in Z . Write the inverse of 4.

Determine whether O on Z defined by a\ O\ b=a^b for all a ,\ b in Z define a binary operation on the given set or not:

A relation R on the set of complex numbers is defined by z_1 R z_2 if and only if (z_1-z_2)/(z_1+z_2) is real Show that R is an equivalence relation.

The operation ** defined by a**b = b^(a) on the set Z = {0,1,2,3,. . . } . Prove that ** is not a binary operation on Z.

On the set Z of integers a binary operation * is defined by a*b= a b+1 for all a ,\ b in Z . Prove that * is not associative on Zdot

For any two complex numbers z_1 and z_2 prove that: |\z_1+z_2|^2=|\z_1|^2+|\z_2|^2+2Re bar z_1 z_2

Construct the composition table for the composition of functions (o) defined on the S={f_1,\ f_2,\ f_3,\ f_4} of four functions from C (the set of all complex numbers) to itself, defined by f_1(z)=z ,\ \ f_2(z)=-z ,\ \ f_3(z)=1/z ,\ \ f_4(z)=-1/z for all z in Cdot

For any two complex number z_1a n d\ z_2 prove that: |z_1+z_2|geq|z_1|-|z_2|

For any two complex number z_1a n d\ z_2 prove that: |z_1-z_2|geq|z_1|-|z_2|