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Interval Notation: A Comprehensive Guide with Examples, Slides of Calculus

The open interval (a,b) represents the set of real numbers between, but not including, a and b. x is greater than a la < xl and x is less lhan b lx <bl.

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Interval
Notation
Subsets of real
numbers
can be represented using
interval notation. Suppose that
0
and b are two real numbers such that
a < b.
Inierval Notation Graph
The open interval (a,b) represents the set
of real numbers between, but not including,
a and b.
(o,b):{"la<x<b}
x is greater
than
a la < xl
and
x is
less
lhan
b lx
<bl.
#]
q (a,b) b
T[e
pareatheses
in
fhe
graph
and
in
interval notalion
indicale
thal a and 1,, ihe
endpoinls, are
excluded from fhe inierval.
The cfosed
interval
la.b)represents
the
set
real numbers
between,
and
including,
a and
fo,bl:{xla<x=b}
x is
greater
lhan or
equal lo a la < xl
and
x is
less than
or equal to
b lx < bl.
#'a [a,h] b
Ihe square brackelr
in
lle
graph
and in
inferval nolalion
indicate
lhsl
a and b, the endpoints,
ere
included
in
lhe interval.
a \a,
r)
The
parenthesis
indicales lhat
a
is excluded
from
the inlerval.
_.+---l+.r (--,bl b
The square
bracket
indicates
that b is included
in the inferyal.
^f
(Jl
h.
The infinite interval (o, *) represents
the
set
of
real
numbers
that are greater
than
a.
(o,*) : {" i
x > a}
The
infinity
symbol
does
not
rePresenl a
real number,
lt
indicales
fhat
lhe interval
exfends
indefinilely
to
the right.
The infinite interval (-oo, bl represents
the
set
of
real
numbers
that are less
than
or equal to b.
(-*, D]
: {x l, = b}
fhe
negalive
infinity
symbol
indicstes
that
fhe interval
extendc iadcfinitely
to
the left.
pf2

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Interval Notation

Subsetsof real

numberscan

be representedusingintervalnotation.Supposethat

and b are two real numberssuchthat a < b.

Inierval Notation Graph

The open interval (a,b) representsthe set

of real numbers between,but not including,

a and b.

( o , b ) : { " l a < x < b }

x is greaterthana la < xl

a n d

x i s l e s sl h a nb l x < b l.

#]

q ( a , b ) b

T[e pareatheses

in fhegraph

a n di n i n t e r v a ln o t a l i o ni n d i c a l e

thal a and1,, ihe endpoinls,are

e x c l u d e df r o mf h ei n i e r v a l.

The cfosedinterval

la.b)representsthe set

real numbers between,and including,a and

f o , b l : { x l a < x = b }

x is greaterlhanor equal lo a la

< xl

a n d

x i s l e s st h a n o r e q u a lt o b l x

< b l.

a

[ a , h ]

b

Ihesquarebrackelrin lle graph

a n di n i n f e r v a ln o l a l i o n i n d i c a t e

lhsla andb, theendpoints, ere

i n c l u d e di n l h ei n t e r v a l.

a

\ a , r )

Theparenthesis indicaleslhat a

is excluded fromthe inlerval.

_.+---l+.r

(--,bl

b

T h es q u a r e b r a c k e ti n d i c a t e s

t h a tb i s i n c l u d e d i n t h ei n f e r y a l.

^ f

( J l

h.

The infinite interval (o, *)

representsthe set of

real numbers

that are greater than a.

( o , * ) :

{ " i x > a }

Theinfinitysymbol does

not

r e P r e s e n la r e a ln u m b e r , l t

indicalesfhatlheinterval

exfendsindefinilely to theright.

The infinite interval (-oo,

bl representsthe set of

real numbers

that are lessthan or equal to b.

( - * , D ] :

{ x l ,

= b }

fhenegaliveinfinitysymbol

indicstesthatfheinterval

extendciadcfinitely to theleft.

Parenthesesand Bracketsin Interval Notation

Parenthesesindicateendpointsthatarenot includedin aninterval.Square

bracketsindicate endpoints that are included in an interval.

Thble 1.5 lists nine possibletypes of intervalsused to describesubsetsof real

numbers.

Table 1.5 Intervals on the Real Number Line

I Let a and b be real numberssuchthat a < b.

:

I

i Interval

I

I Notation Set-BuilderNotation

: Graph

i @ , b )

f r t

I Q. D I

i

,,

La b)

r l

\ u , D )

{ t l o <

x < b }

t a b

t r l , r = x = b \

'

i

I

'.

a b

{ -. l n < x < b }

, , o b

{ " l n < x = b l

r

a b

( a , o o )

{

x l x > a \

l #. r

L '

i

[ 4 ,

o o )

{ x l x

a }

\

' o - , u )

t. t l r \ u - i

r t

i

( - - , o o

) { , r l x

i s a r e a l n u m b e r } o r R

1

(setof all real numbers) ,

x ' h l

' I

\

.. r

t. Y l. t

l 7 i

.l