Among the statements (S1) : The set z C - - i :| z |=1 . and z - i z + i is purely real contains exactly two elements, and (S2) : The set z C - -1 :| z |=1 . and z -1 z +1 is purely imaginary contains infinitely many elements.

Step-by-step solution

aligned & S ₁:| z |=1, z - i z + i = z + i z - i & ( z - i )( z - i )=( z + i )( z + i ) & | z |^2- i ( z + z )-1=| z |^2+ i ( z + z )-1 & i ( z + z )=0 & z + z =2 =0 =0 & z =0+0 i ,| z | 1 & ~S ₁: z -1 z +1 + z -1 z +1 =0 & ( z -1)( z +1)+( z +1)( z -1)=0 & | z |^2+( z - z )-1+| z |^2+( z - z )-1=0 & | z |^2=1 aligned