1.1 BASIC DEFINITIONS AND NOTATION 11Subset. Earlier we encountered a relationship of containment between an
object and a set. Denoted E, this relationship symbolizes membership, or
elementhood, of the object as one of the elements in a set. There is also a
concept of containment between sets, known as the subset relationship.
As we did for set equality, we give an informal description of this concept
now, with the formal definition provided in Definition l(b), Article 4.1.
REMARK 3 Let A and B be sets. We regard the statement A is a subset of
B, denoted A E B, to mean that every element of A is also an element
of B. We write A $ B to denote that A is not a subset of B. FinalIy, we
define B is a superset of A to mean A E B.
We observed earlier that the set D of the eight states whose names begin
with the letter M is not the same as the set S = (Maine}. A correct relation-
ship is S s D.
EXAMPLE 3 (a) Find all subset relationships among the sets H = (1,2,
3), N = (2,4,6, 8, lo), and P = (1, 2, 3,... , 9, 10).
(b) Find all subset relationships among the sets T = {2,4,6,.. .},
V = (4,8, 12;.. .}, and W = (... , -8, -4,0,4,8,.. .).
Solution (a) Clearly H G P and N sz P, as you can easily see by checking,
one by one, that all the individual elements of H and N are elements of
P. Don't be misled by the representation of P, which contains ten ele-
ments. Also, P $ H and P $ N (can you explain precisely why this
is true?), whereas neither H nor N is a subset of the other (why?).
(b) This problem is more difficult than (a) because the sets involved
are infinite. Intuition about the connections among the underlying de-
scriptions of these sets (e.g., V is the set of all positive integral multiples
of 4) should lead to the conclusions V G T, V E W, T $ V, W $ V, and
neither T nor W is a subset of the other. You should formulate an argu-
ment justifying the latter statement.
The five number sets designated earlier by name satisfy the subset re-
lationships N c Z, Z c Q, Q c R, and R E C. The subset relation, like
set theoretic equality, enjoys the reflexive and transitive properties. Put
more directly, ev&y set is a subset of itself, and for any sets A, B, and C, if
A E B and B c C, then A c C. The subset relation is not symmetric (try
to formulate precisely what this statement means). Furthermore, Example
3 illustrates the fact that, given two sets, it may very well be that neither
is a subset of the other. The reader will note that the transitivity of the subset
relation has consequences for the examples in the previous paragraph,
namely, a number of additional subset relationships among the five sets listed
there are implied. You should write down as many of those relationships
as possible.