8 SETS Chapter 1known as intervals, which we will frequently encounter. Intervals are de-
scribed by widely used notation, which we will soon introduce. Before do-
ing so, we give a definition of interval that provides our first example of an
abstract mathematical definition of the type we will often work with later
in the text.DEFINITION 2
A set I, all of whose elements are real numbers, is called an interval if and only if,
whenever a and b are elements of 1 and c is a real number with a < c < b, then
c E I.Intervals are characterized among other sets of real numbers by the
property of containing any number between two of its members. All in-
tervals must do this and intervals are the only sets of real numbers that
do this. In particular, any set of real numbers such as (0, 1,2), or Z or Q,
which fails to have this property, is not an interval. Intervals are easy to
recognize; indeed, we will prove in Chapter 9 that every interval in R has
one of nine forms.DEFINITION 3
Nine types of intervals are described by the following terminology and notation, in
which a and b denote real numbers:
I. [x E Rla 5 x 5 b), a closed and boundedinterval. denoted [a, b],- (X E Rla < x < b), an open and bounded interval, denoted (a, b),
- (x E R(a 5 x < b), a closed-open and bounded interval, denoted [a, b),
- (x E la < x 5 bj., an open-closed and bounded interval, denoted (a, b],
- (x E R(a I x), a closed and unbounded above interval, denoted [a, a),
6. (x E la < x), an open and unbounded above interval, denoted (a, a),
7. {x E R(X I b), a closed and unbounded below interval, denoted (- oo, b],
8. {x E R 1 x < bf, an open and unbounded below interval, denoted (- oo, b),
9. R itself is an interval and is sometimes denoted (-a, a).
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Intervals arise in a large variety of mathematical contexts and in particular
are involved in the statement of numerous theorems of calculus. A familiar
application of interval notation at a more elementary level is in expressing
the solution set to inequalities encountered in elementary algebra.
EXAMPLE 1 Assuming that the universal set is R, solve the following in-
equalities and express each solution set in interval notation:
(a) 7x-9 5 16
(b) 12x+31<5
(c) 2x2 + x -^28 5 0