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` ?

An operation ** on the set of all complex numbers CC is defined by z_(1)**z_(2)=sqrt(z_(1)z_(2)) for all z_(1),z_(2)inCC . Is ** a binary operation on CC ?

A relation R on the set of complex number is defined by z_1 R z_2 iff (z_1 - z_2)/(z_1+z_2) is real ,show that R is an equivalence relation.

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

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

Let the set S={f_(1),f_(2),f_(3),f_(4)} of four functions from CC (the set of all complex numbers) to itself, defined by f_(1)(z)=z,f_(2)(z)=-z,f_(3)(z)=(1)/(z)andf_(4)(z)=-(1)/(z) for all zinCC Construct the composition table for the composition of functions (@) defined on the set S. value of f_(2)@f_(1)(z) is--

Let the set S={f_(1),f_(2),f_(3),f_(4)} of four functions from CC (the set of all complex numbers) to itself, defined by f_(1)(z)=z,f_(2)(z)=-z,f_(3)(z)=(1)/(z)andf_(4)(z)=-(1)/(z) for all zinCC Construct the composition table for the composition of functions (@) defined on the set S. Value of f_(2)@f_(4)(z) is---

Let the set S={f_(1),f_(2),f_(3),f_(4)} of four functions from CC (the set of all complex numbers) to itself, defined by f_(1)(z)=z,f_(2)(z)=-z,f_(3)(z)=(1)/(z)andf_(4)(z)=-(1)/(z) for all zinCC Construct the composition table for the composition of functions (@) defined on the set S. Value of f_(4)@f_(1)(z) is ---

For any two complex numbers z_1 and z_2 , , prove that Re (z_1 z_2) = Re z_1 Re z_2 – Imz_1 Imz_2

For all complex niumbers z_1, z_2 satisfying absz_1=12 and abs(z_2-3-4i)=5 , the minimum value of abs(z_1-z_2) is

If z_1a n dz_2 are complex numbers and u=sqrt(z_1z_2) , then prove that |z_1|+|z_2|=|(z_1+z_2)/2+u|+|(z_1+z_2)/2-u|