1.2 OPERATIONS ON SETS 15- Given the following collections of sets, find in each of parts (a), (b), and (c) all
relationships of equality, subset, and proper subset existing between pairs of them:
(a) A={-1, ~,~),B=(-~,~),C={~ERIX~-~X~-X+~=O),D=[-~,~].
(b) A = {0,0, I), B = (0, {a)), C = [O, 11, D = {{0,1), {O), {1),0, {a)
(c) A= {xENIIx~ s4}, B= {-4, -3, -2, -1,0, 1,2,3,4),C= {xEZ~)~ 1 < 5). - (a) Considering the definition of interval given in Article 1.1, explain precisely
why the set (0, 1,2) is @ an interval.
*(b) What statement can be made about the subsets Z and Q of R, based on Defi-
nition 2 and the assertion (from the paragraph following Definition 2) that Z
and Q are not intervals? - Considering the "definition" of subset given in Article 1.1 (cf., Remark 3), discuss
the pros and cons of the statement 0 E {1,2,3), that is, can you see arguments
for both the truth and falsehood of this statement? What about 521 E 0? What
about 0 G A, where A is any set? - Throughout this problem, assume the statement "0 c A for any set A" is true:
(a) Calculate 9(S) for:
(i) S={l,2,3} *(ii) S = {a, b, c, d )
(iii) s = (^0) (W S={0)
(v) s = (0, (0)) (vi) S = P(T) where T = (1,2)
(b) Can you list all the elements of P((1,2,3,.. .))? List ten such elements.
(c) Can you give an example of a finite set X such that #(X) is infinite?
"10. Suppose U were a truly universal set; that is, U contains all objects. Then, in
particular, U would contain itself as an element, that is, U E U. This is an unusual
situation since most sets that one encounters do not contain themselves as an ele-
ment (e.g., the set X of all students in a mathematics class is not a student in that
class; that is, X # X in this case.) Now consider the "set" A of all sets that are not
elements of themselves; that is, A = {Y I Y q! Y). Discuss whether A E A or A $ A.
Operations on Sets
As stated earlier, sets like Z or R or (1,2, 31, consisting of numbers, are of
greater mathematical interest than most other sets, such as the set of all
names in a telephone book. One reason is that numbers are mathematical
objects; that is, numbers can be combined in various mathematically in-
teresting ways, by means of algebtaic operations, to yield other numbers.
Among the operations on real numbers that are familiar are addition, sub-
traction, multiplication, and division. You should also know that these
types of operations may satisfy certain well-known properties, such as
commutativity, associativity, and distributivity. For example, addition and
multiplication over the real numbers both satisfy the first two properties,
while multiplication distributes over addition. On the other hand, sub-
traction over real numbers is neither commutative nor associative.