Page 14 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 1] NUMBERS 5

POINT SETS, INTERVALS
A set of points (real numbers) located on the real axis is called a one-dimensional point set.
The set of points x such that a @ x @ b is called a closed interval and is denoted by ½a; bŠ. The set
a < x < b is called an open interval,denoted by ða; bÞ. The sets a < x @ b and a @ x < b, denoted by
ða; bŠ and ½a; bÞ, respectively, are called half open or half closed intervals.
The symbol x, which can represent any number or point of a set, is called a variable. The given
numbers a or b are called constants.
Letters were introduced to construct algebraic formulas around 1600. Not long thereafter, the
philosopher-mathematician Rene Descartes suggested that the letters at the end of the alphabet be used
to represent variables and those at the beginning to represent constants. This was such a good idea that
it remains the custom.

EXAMPLE. The set of all x such that jxj < 4, i.e., 4 < x < 4, is represented by ð 4; 4Þ,an open interval.
The set x > a can also be represented by a < x < 1. Such a set is called an infinite or unbounded
interval. Similarly, 1 < x < 1 represents all real numbers x.

COUNTABILITY
A set is called countable or denumerable if its elements can be placed in 1-1 correspondence with the
natural numbers.

EXAMPLE. The even natural numbers 2; 4; 6; 8; ... is a countable set because of the 1-1 correspondence shown.
Given set 2 4 6 8 ...
l lll
Natural numbers 1 2 3 4 ...
A set is infinite if it can be placed in 1-1 correspondence with a subset of itself. An infinite set which
is countable is called countable infinite.
The set of rational numbers is countable infinite, while the set of irrational numbers or all real
numbers is non-countably infinite (see Problems 1.17 through 1.20).
The number of elements in a set is called its cardinal number. A set which is countably infinite is
assigned the cardinal number F o (the Hebrew letter aleph-null). The set of real numbers (or any sets
which can be placed into 1-1 correspondence with this set) is given the cardinal number C, called the
cardinality of the continuuum.

NEIGHBORHOODS
The set of all points x such that jx aj < where > 0, is called a neighborhood of the point a.
The set of all points x such that 0 < jx aj < in which x ¼ a is excluded, is called a deleted
neighborhood of a or an open ball of radius about a.

LIMIT POINTS

A limit point, point of accumulation,or cluster point of a set of numbers is a number l such that
every deleted neighborhood of l contains members of the set; that is, no matter how small the radius of
a ball about l there are points of the set within it. In other words for any > 0, however small, we can
always find a member x of the set which is not equal to l but which is such that jx lj < . By
considering smaller and smaller values of we see that there must be infinitely many such values of x.
A finite set cannot have a limit point. An infinite set may or may not have a limit point. Thus the
natural numbers have no limit point while the set of rational numbers has infinitely many limit points.