Advanced High-School Mathematics

188 CHAPTER 4 Abstract Algebra

Z⊆Q⊆R⊆C.

As a more geometrical sort of example, let us consider the setR^3 of
all points in Cartesian 3-dimensional space. There are certain naturally
defined subsets ofR^3 , thelinesand theplanes. Thus, if Π is a plane
inR^3 , and if Lis a line contained in Π, then of course we may write
eitherL⊂Π⊂ R^3 orL⊆Π⊆ R^3. Note, of course, thatR^3 has far
more subsets that just the subsets of lines and planes!

One more example might be instructive here. First of all, if Ais a
finite set, we shall denote by|A|the number of elements inA. We often
call|A|thecardinalityororderof the setA. Now consider the finite
setS = { 1 , 2 , 3 , ..., 8 }(and so|S|= 8) and ask how many subsets
(includingS and the empty set∅) are contained inS. As you might
remember, there are 2^8 such subsets, and this can be shown in at least
two ways. The most direct way of seeing this is to form subsets ofS
by the following process:

1 2 3 4 5 6 7 8

yes

or no

yes

or no

yes

or no

yes

or no

yes

or no

yes

or no

yes

or no

yes

or no

where in the above table, a subset if formed by a sequence ofyes’s or
no’s according as to whether or not the corresponding element is in
the subset. Therefore, the subset{ 3 , 6 , 7 , 8 }would correspond to the
sequence

(no, no, yes, no, no, yes, yes, yes).

This makes it already clear that since for each element there are two
choices (“yes” or “no”), then there must be

2 × 2 × 2 × 2 × 2 × 2 × 2 ×2 = 2^8

possibilities in all.

Another way to count the subsets of the above set is to do this: