Sketch the sets & Determine whether they are open,closed, or neither - bounded o

Question

Sketch the sets & Determine whether they are open,closed, or neither - bounded or unbounded - connected or not connected

(a) |z+3| < 2

(b) |Im(z)| < 1

(c) 0<|z1|<2

(d) |z1|+|z+1| = 2

(e) |z1|+|z+1| < 3

(f) |z|Re(z)+1

*** NOTE: The work has been done, no need to sketch or find the sets except for part (e). ***

I just need for EACH (a),(b),(c),(d),(e),(f) is it open or closed or neither ; bounded or unbounded ; connected or not connected

(a) Interior of the circle with center -3 & radius 2

(b) the dotted lines -1 < y < 1

(c) Interior of the circle center 1 and radius 2 ; points in the interior of circle excluding the center at 1

(d) after simplifying got y=0 so the set is the x axis (line)

*** (e) after simplifying got y^2= 1/8 (not sure about this sketch - NEED FULL SOLUTION HERE)

(f) parabola with vertex (-1/2,0) facing to the right crosses y axis at +1 and -1.

Explanation / Answer

a.the set is open as it does not contains its boundry.it is connected since disk is connected .it is bounded

b.the set is unbounded,connected and open.it is both open and closed

c.the set is bounded,open and connected

d.the set is real line..so it is connected,unbounded.it is both open as well as closed.as it is complement of empty set.

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