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23 Maths -- Sets, Real Number System and Logic

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Real Numbers- Axioms

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Hello Readers!!
Click Here for a quick overview of Real numbers before proceeding with this note!!! :) 

Field Axioms

1.Addition axiom



























i. Closure Property 

:

The sum of real number is always real value.
i.e., If {a, b} ϵ R then a+b ϵ R

ii. Commutative Property

:

The sum of two real numbers is the same whatever order they are added.
i.e., If {a, b} ϵ R then a + b = b + a

iii. Associative Property

:

The sum of any two of the three real numbers with the third will be the same.
i.e., if {a, b, c} ϵ R then, a+(b+c)=(a+b)+ c

iv. Additive Identity 

:

If a ϵ R, then a+ 0 = 0 + a =a
Here, 0 is the additive identity.

v. Additive Inverse

:

For every a ϵ R, there exists -a ϵ R such that:
a+(-a) = (-a) +a = 0
Here, -a is the additive inverse of a

2. Multiplicative Axiom
































i. Closure Property 

:

The product of real number is always real value.
i.e., If {a, b} ϵ R then a.b ϵ R

ii. Commutative Property

:

The product of two real numbers is the same whatever order they are multiplied.
i.e., If {a, b} ϵ R then a.b = b.a

iii. Associative Property

:

The product of any two of the three real numbers with the third will be the same.
i.e., if {a, b, c} ϵ R then, a.(bc)=(ab). c

iv. Multiplicative Identity 

:

If a ϵ R, then a.1 = 1.a =a
Here, 1 is the multiplicative identity.

v. Multiplicative Inverse

:

For every aϵ R(a≠0), there exists a-1 ϵ R
such that:
a. a-1 = a-1. a = 1
Here, a-1 is the multiplicative inverse of a

vi. Distributive Property

:

If {a, b, c} ϵ R, then
a(b+c)=ab+bc


Order Axioms
i. If {a, b} ϵ R such that a>0 and b>0 then ab>0.
ii. If {a, b} ϵ R then one and only one of the following relation holds
                            a<b,     a=b,     a>b
iii. If {a, b, c} ϵ R such that a>b and b>c then a>c.(same is the case for '<')
iv. If {a, b, c} ϵ R such that a>b then a+c > b+c.(same is the case for '<')
v. {a, b, c} ϵ R 
    →If a>b            then             ac>bc            when c>0
    →If a>b            then             ac<bc            when c<0
    →If a>b            then             a/c>b/c        when c>0 
    →If a>b            then             a/c<b/c        when c<0 
                                    (same is the case for '<')
vi. If {a, b} ϵ R such that a<b, then there exists a real number such that: a<c<b 


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