1.3 ALGEBRAIC PROPERTIES OF SETS 37standing of what the issues are, based on the following discussion, which
is essentially an answer to the aforementioned Exercise 7.
Using Remark 3 of Article 1.1 (defining X E A by "every element of X
is also an element of A"), we could drgue (fallaciously) that (a (1,2)
(letting A = (1,2) for concreteness) because there are no elements in (a
and thus no elements of 0 that are also elements of {1,2}. On the other
hand, we could reason (correctly, it turns out) that, for a set X to fail to
be a subset of A, there must be an element of X that is not an element of A.
Since X = 0 has no elements, this is impossible, so that 0 cannot fail to
be a subset of A; that is, 0 G A.
As for the second principle, we need only emphasize that, by definition,
0 is the set containing no elements, so that 0 E 0 must be false. If it
were true, then, for example, {2,4,6, 8, 10) u (21 would equal {2,4,6, 8,
10, 0) rather than the correct (2,4,6,8, 10) (recall this question from
Example 2, Article 1.2). Other important facts to note on this basis are that
0 # (0) (the latter set has one element, the former has none) and (a # (0)
(even though zero does equal the number of elements in (a). Furthermore,
whereas (ZI E (ZI is false, (21 E {(a) is true. Think of /a as an empty box
and ((a) as a box containing an empty box. The former is empty; the lat-
ter is not empty for it contains something, an empty box!
With these principles in mind, you should be able to convince yourself
that the answers to the three questions posed at the start of this section are:- False (if (ZI were an element of 0, it would be true)
- False (since (0) # (a u (0))
- True (since 0 E X for any set X, then (a E 9(X) for any set X)
We stated earlier that (a and U are at the "opposite ends of the spectrum."
There are a number of senses in which this statement is accurate. One is
that the two sets are complementary, that is, 0' = U and U' = 0. Another
is an extension of a fact we just discussed. Just as 0 is a subset of any set,
so is U a superset of any set (i.e., any set is a subset of U). Finally, the
empty set equals the intersection of any set with its complement, whereas
a universal set equals the union of any two such sets.
-- -.
Exercises
- Make a list of possible theorems of set theory suggested by the answers calculated
in Exercises 1 through 5 of Article 1.2. - Let U = (1,2,3,... ,9, 10). Find specific subsets A, B, C, and/or X of U that
contradict (i.e., disprove) the conjectures listed here. For any subsets A, B, C, and
X of U:. '4
(a) A - (B-.C)=(A - B) - C *(b) (A - B)'= A - B1 %
(c) A-(B1uC)=(AuB)-C (d) AAB=AuB