Describe geometrically the following subsets of C.
(i) `{ z in C||z-1+i|=1}`
(ii) `{z in C|| z+i| le 3|`

Explanation

Let (i) `S={ z in C||z-1+i|=1}`
If we `z=(x,y)` then
`S={(x,y) in R^(2)||x+ iy-1+i|=1}`
`={(x,y), in R^(2)|| x+i(y+1)| le 3}`
`={(x, y) in R^(2)||(x-1)^(2)+(y+1)^(2)=1}`
Hence S is circle with centre (1,-1) and radius 1 unit.
(ii) Let `S'={ z in C|| z+i| le 3}`
then `S={(x,y) in R^(2)|| x+ iy+i | le 3}`
`={(x,y) in R^(2) || x+i(y+1) | le 3}`
`={(x,y) in R^(2) || x^(2)+(y+1)^(2) le 9}`
Hence S' is the closed circular disc with centre at (0,-1) and radius 3 units.