Providing an full conceptual notes for master level students of subject Mathematics. These notes include basic information of subject Real Analysis. For vast Concept and detailed learning these are beneficial for students.
Mathematical analysis module 1 Real numbers PYQ solved
Real Analysis 2 chapter 8 ( Exercise 8.1)
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UNIVERSITY OF THE PUNJAB, LAHORE
MATH301 (MATH301)
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M.Se
Matk REALANALYSIS
CH#
Wumbes Systems.
Real Anaysis:
Tebhanch f'malhemathes
shich dals wih sudy frpesta
nubers and the funechons fpm et ef
ralrumbers to set
Aea
Ten numbers.
Real Nuwnbers
sfem tnumbexs T from- to +iSCalled
fead number System denoted by R=]-a,qL
Infaet,e number Sustem is a linenumbers
-g
Flous Chont f Number Systems:
R-SetofealNumbers.
Q-Set raienal Numbers
-Set e Rationa Nanbers
Z Set Tateye Ys
W Set a whoe rumbes
N Set of Natwra Nawbera
Nambers wih maqinary
Set d all real numbers is a set Compex
Past Zeyos R -C-{a,b)} aeR ad bro
, Paopertes oReal Number ystem:
) Field Propexties:
w.rt, Addition .T.t. MuHipliealian
A: Closur Propeat M,: Closure Propesty
Y a,b e R Va,beR
a + b eR a.be R
A: Commutative fropert M:ommuthtive Popety
Y a,beR a,be
a b = bta a.b = b a
A Associahive Popes M:Amociathe Pmpen:
a,b,ceR
a+(btc) =(a+b)+tc ab-c)= ab).c
AIelentiExistance: My: Identi Enistance:
VaeR 1 oeR YaeR 1e R
such that
Sueh thot
a+oe o+a = Qa
a.L=1 a 2 a
"'o Called Additie iletity
i
Ts called
A: Inerse Existance: muthpicahie idenl.
M: Invere Eristance:
Y aeR -atR VaeR 3 LeR
Such hat Such That
a+-a) =-a +a=0
-a" is called Additie
a.
a
=1
inverse a" 6 Caled mulhiplèahive
inverse a"
Disthibutie Propest:
V a,b,cE R
a.Cbte) = a-bta.c
a+b)C = a.c+b.c Left Distaibutive La
A
Set that satisfres Riht Distibuhe Lud
all te
above 11
Field ad is
denoted as
propesties is clue
Set of rationa) (F+,
(F+,)-)
numbers
numbe rs, Set of rea) numbers avd Set
are 3
fields. ofamplex
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