Discuss: Is a convex set, defined in Problem, necessarily connected?ProblemA set S is said...

Discuss: Is a convex set, defined in Problem, necessarily connected?

Problem

A set S is said to be convex if each pair of points P and Q in S can be joined by a line segment  such that every point on the line segment also lies in S. Determine which of the sets S in the complex plane defined by the following conditions are convex.

(a) |z − 2 + i| < 3

(b) 1 < |z| < 2

(c) x > 2, y ≤ −1

(d) y < x2

(e) Re(z) ≤5

(f) Re(z) ≠ 0