Page 138 - Discrete Mathematics and Its Applications

2.1 Sets 117

(Note that some people do not consider 0 a natural number, so be careful to check how the term
natural numbers is used when you read other books.)
Recall the notation for intervals of real numbers. When a and b are real numbers with
a< b, we write

[a, b]={x | a ≤ x ≤ b}

[a, b) ={x | a ≤ x< b}
(a, b]={x | a< x ≤ b}
(a, b) ={x | a< x < b}

Note that [a, b] is called the closed interval from a to b and (a, b) is called the open interval
from a to b.
Sets can have other sets as members, as Example 5 illustrates.
EXAMPLE 5 The set {N, Z, Q, R} is a set containing four elements, each of which is a set. The four elements
of this set are N, the set of natural numbers; Z, the set of integers; Q, the set of rational numbers;
and R, the set of real numbers. ▲

Remark: Note that the concept of a datatype, or type, in computer science is built upon the
concept of a set. In particular, a datatype or type is the name of a set, together with a set of
operations that can be performed on objects from that set. For example, boolean is the name of
the set {0, 1} together with operators on one or more elements of this set, such as AND, OR,
and NOT.

Because many mathematical statements assert that two differently specified collections of
objects are really the same set, we need to understand what it means for two sets to be equal.

DEFINITION 2 Two sets are equal if and only if they have the same elements. Therefore, if A and B are sets,
then A and B are equal if and only if ∀x(x ∈ A ↔ x ∈ B). We write A = B if A and B are
equal sets.

EXAMPLE 6 The sets {1, 3, 5} and {3, 5, 1} are equal, because they have the same elements. Note that the
order in which the elements of a set are listed does not matter. Note also that it does not matter
if an element of a set is listed more than once, so {1, 3, 3, 3, 5, 5, 5, 5} is the same as the set
{1, 3, 5} because they have the same elements. ▲

GEORG CANTOR (1845–1918) Georg Cantor was born in St. Petersburg, Russia, where his father was a
successful merchant. Cantor developed his interest in mathematics in his teens. He began his university studies
in Zurich in 1862, but when his father died he left Zurich. He continued his university studies at the University
of Berlin in 1863, where he studied under the eminent mathematicians Weierstrass, Kummer, and Kronecker.
He received his doctor’s degree in 1867, after having written a dissertation on number theory. Cantor assumed
a position at the University of Halle in 1869, where he continued working until his death.
Cantor is considered the founder of set theory. His contributions in this area include the discovery that the
set of real numbers is uncountable. He is also noted for his many important contributions to analysis. Cantor
also was interested in philosophy and wrote papers relating his theory of sets with metaphysics.
Cantor married in 1874 and had five children. His melancholy temperament was balanced by his wife’s happy disposition.
Although he received a large inheritance from his father, he was poorly paid as a professor. To mitigate this, he tried to obtain a
better-paying position at the University of Berlin. His appointment there was blocked by Kronecker, who did not agree with Cantor’s
views on set theory. Cantor suffered from mental illness throughout the later years of his life. He died in 1918 from a heart attack.