A function is bounded above if the function pos-
sesses an upper bound (but not necessarily a lower
bound), and bounded below if it possesses a lower
bound (but not necessarily an upper bound). For exam-
ple, f(x) = x2is bounded below by the value L= 0,
since all output values for this function are greater than
or equal to zero. The
EXTREME
-
VALUE THEOREM
ensures that every
CONTINUOUS FUNCTION
defined on a
closed
INTERVAL
is bounded.
A set of numbers is bounded above if every number
in the set is less than or equal to some value M,
bounded below if every number in the set is greater
than or equal to some value L, and bounded if it is
both bounded above and bounded below. For example,
the set S= {0.6, 0.66, 0.666, …} is bounded below by
0.6 and bounded above by 1. The smallest possible
upper bound for a set is called the least upper bound,
and the largest lower bound, the greatest lower bound.
The set Shas 2/3 as its least upper bound and 0.6 as its
greatest lower bound.
The
REAL NUMBERS
have the property that any
subset Sof them that is bounded above possesses a
least upper bound, and, similarly, any subset that is
bounded below possesses a greatest lowest bound.
This is a key property that shows that no numbers are
“missing” from the real number line. (See D
EDEKIND
CUT
.) This is not true of the set of rational numbers,
for instance. The set of all rationals whose square is
less than 2, for example, is bounded above, by 3/2 for
example, but possesses no least upper bound in the set
of rationals: the square root of two is “missing” from
the set of rationals.
A sequence is bounded if, as a set of numbers, it is
bounded. A geometric figure in the plane is bounded if
it can be enclosed in a rectangle of finite area. For
example, a
CIRCLE
is bounded but a
HALF
-
PLANE
is not.
Bourbaki, Nicolas Taking the name of a junior
Napoleonic officer, a group of French mathematicians of
the 1930s adopted the pseudonym of Nicolas Bourbaki
to publish a series of books, all under the title Éléments
de mathématiques (Elements of mathematics), that
attempt to present a complete, definitive, and utterly rig-
orous account of all modern mathematical knowledge.
This project continues today. Contributors to the work
remain anonymous and change over the years. To date,
over 40 volumes of work have been produced.
The material presented through Bourbaki is austere
and abstract. The goal of the founding work was to
develop all of mathematics on the axioms of
SET THE
-
ORY
and to maintain the axiomatic approach as new
concepts are introduced.
The work, devoid of narrative and motivational con-
text, is difficult to read and not suitable for use as text-
books. During the 1950s and 1960s, however, there were
often no graduate-level texts in the developing new fields,
and the volumes of Bourbaki were the only sources of
reference. It is unlikely that today a graduate student in
mathematics would consult the work of Bourbaki.
brachistochrone See
CYCLOID
.
brackets Any pair of symbols, such as parentheses ( )
or braces { }, that are used in an arithmetic or an alge-
braic expression to indicate that the quantity between
them is to be evaluated first, or treated as a single unit
in the evaluation of the whole, are called brackets. For
example, in the expression (2 + 3) ×4, the parentheses
indicate that we are required to first calculate 2 + 3 = 5
and then multiply this result by 4. In complicated
expressions, more than one type of bracket may be
used in the same equation. For instance, the expression
3{2 + 8[2(x+ 3) – 5(x– 2)]} is a little easier to read
than 3(2 + 8(2(x+ 3) – 5(x– 2))).
Before the advent of the printing press in the 15th
century, the
VINCULUM
was used to indicate the order
of operations. Italian algebraist R
AFAEL
B
OMBELLI
(1526–1572) was one of the first scholars to use paren-
theses in a printed algebraic equation, but it was not
until the early 1700s, thanks chiefly to the influence of
L
EONHARD
E
ULER
, G
OTTFRIED
W
ILHELM
L
EIBNIZ
, and
members of the B
ERNOULLI FAMILY
, that their use in
mathematics became standard.
Angle brackets <> are typically only used to list the
components of a
VECTOR
or a finite
SEQUENCE
. Matters
are a little confusing, for in the theory of quantum
mechanics, angle brackets are used to indicate the
DOT
PRODUCT
of two vectors (and not the vectors them-
selves). The left angle bracket “<” is called a “bra” and
the right angle bracket “>” a “ket.”
In
SET THEORY
, braces { } are used to list the ele-
ments of a set. Sometimes the elements of a sequence
are listed inside a set of braces.
50 Bourbaki, Nicolas