Among the statements (S1): The set ${z\in\mathbb{C}\setminus{-i}:\ |z|=1\ \text{ and }\ \dfrac{z-i}{z+i}\ \text{is purely real}}$ contains exactly two elements and (S2): The set ${z\in\mathbb{C}\setminus{-1}:\ |z|=1\ \text{ and }\ \dfrac{z-1}{z+1}\ \text{is purely imaginary}}$ contains infinitely many elements.