Real analysis introduction, lecture 1, Real Analysis

Course title: Real Analysis I
course code: MTH 201, 4 credits
Course conducted by Sapla shirin(SS)

Date :03.03.2011

Mathematical analysis studies concepts related in some way to real numbers. So we begin our study of real analysis with a discussion of the real number.

AXIOMS OF REAL NUMBERS
We assume there exists a non empty set R of objects called real numbers which satisfy the ten axioms listed bellow-

The axioms fall in a natural way into three groups which we refer to as

1. The field axioms
2. The Order axioms
3. The completeness axioms

THE FIELD AXIOMS
Along with the set of real number we assume the existence of two operations called addition and multiplication such that for every pair of real numbers x and y the sum x+y uniquely determined by x and y satisfying the following axioms

A1: Closure law of addition, for all a,b ϵ R , a + b ϵ R 

A2: Commutative law of addition,  for all a,b ϵ R , a + b = b+a

A3: Associative law of addition, for all a,b,c  ϵ R , a +( b+c)=(a+b)+c

A4: Existence of additive Identity,  for all a ϵ R , there exists 0 ϵ R such that a+0=0+a=a

A5: Existence of additive inverse, for all a ϵ R , there exists -a ϵ R such that a+(-a)=(-a)+a=0

********(To be continued) ********