Complex Numbers Part 1

For $S_{1}, |z-1| \le \sqrt{2}$,....(i) $z$ lies on and inside the circle of radius $\sqrt{2}$ units and centre $(1,0)$. For $S_{2}$, let $z=x+iy$ Now, $(1-i)(z)=(1-i)(x+iy)$ $=x+iy-ix+y$ $=(x+y)+i(y-x)$ $\therefore \operatorname{Re}[(1-i)z]=(x+y)$, which is greater than or equal to one. i.e., $x+y \ge 1$...(ii) Also, for $S_{3}$, Let $z=x+iy$ $\therefore \operatorname{Im}(z)=y$, which is less than or equal to one. i.e., $y \le 1$ ....(iii) Concept Draw the graph of Eqs. (i), (ii) and (iii) and then select the common region bounded by Eqs. (i), (ii) and (iii) for $S_{1} \cap S_{2} \cap S_{3}$.$\therefore S_{1} \cap S_{2} \cap S_{3}$ has infinitely many elements.