Page 2 - Real Analysis I-e content (1)_Neat

Real analysis-I

Introduction:

The rational and irrational numbers together constitute real numbers.

Set notation:

Let S denote a set. The notation means the object is in the set S is not in S .

A set S is said to be a subset of T , we write if every object in S also in T . A set is called non empty
if it contains atleast one object. A non empty set R of objects real numbers which satisfy the below
ten axioms.

Field Axioms:

The set R of real numbers we assume the existence of 2 operations called addition and mulitiplication
for every pair of real numbers x & y we have x+y & xy

Axiom 1: x  y  y  x, xy  yx (commutative law)

 x 
Axiom 2: x  y  z  y  xy  yx (associative law)
 z,
 xy
Axiom 3: y  z  xz (distributive law)
x
Axiom 4: Given any 2 real numbers x & y there exists a real number z such that
x  y  y  z  y  x & x  x is denoted by 0.

0
x
Axiom 5:There exists at least one real number  there exists a real number z such that
xz  y  z  y x /
x
The number x / is denoted by 1 is independent of x
The order axioms:

Assume the existence of less than which establishes an ordering among the real numbers.
Axiom 6:Exactly one of the relations x  y  x  y & x  y holds.

Axiom 7: If x  y then for every z we have x  z  y  z

Axiom 8: If x 0 & y  0 then xy  0

Axiom 9: If x  y & y  z then x 
z
Axiom 10: Completeness axiom:

Every nonempty set S of real numbers which is bounded above has supremum.

Ie., there is real number b such that b  sup S