Complex Numbers Part 1

For S1,|z−1|≤√2,....(i)
z lies on and inside the circle of radius √2 units and centre (1,0).
For S2, let z=x+iy
Now, (1−i)(z)=(1−i)(x+iy)
=x+iy−ix+y
=(x+y)+i(y−x)
∴Re[(1−i)z]=(x+y), which is greater than or equal to one.
i.e., x+y≥1...(ii)
Also, for S3,
Let z=x+iy
∴Im(z)=y, which is less than or equal to one.
i.e., y≤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 S1∩S2∩S3.∴S1∩S2∩S3 has infinitely many elements.