Topology of Plane Sets of Points(b) A 6 (B 6 C) = (A 6 B) 6 C
(c) An (B 6 C) = (An B) 6 (An 0)
( d) A 6 0 = A, A 6 A = 0- Prove De Morgan's laws for arbitrary unions and intersections.
81- A ring of sets is a nonempty collection G of sets such that if A E G and
B E G, then A 6 B and An B are in G. Show that G must contain
0, AU B, and A - B.
2.4 SETS OF COMPLEX NUMBERS
In this .. section we give a number of examples of sets of complex numbers
(or equivalently, of s~ts of points in the complex plane) defined in terms
of certain simple equalities or ineq~alities. Many of these sets correspond
to well-known geometric loci. Henceforth a point in the plane, a vector
from the origin to that point, and the associated complex number will be
denoted by the same symbol.
L Straight line. We have seen in Section 1.3 that the equation. ' '
Az+Az+C = 0 (2.4-1)
where A # 0 is a complex constant, C a real constant, and z a complex
variable, represents a· straight line in. the complex plane (see also Exer-
cises 1.2, problem 24). This straight line L passes through the point
z 0 ~ -C /2A = -OA/2IAl^2 , and it is perpendicular to the directiori of
the vector A (Fig. 2.1). In fact, if we let A = a+ i(J, z = x + iy, the. equation (2.4-1) becomes
2ax + 2(3y + G = 0 ' ·
y
L
Ax
Fig. 2.1