Question

दिखाईए योज्य द्विधारी सक्रिया के साथ सभी पूर्णांको का समुच्चय I अनंत अबोली समूह होता है​

Answer 1

SOLUTION

TO PROVE

The set of integers forms an abelian group under addition

PROOF

The set of integers is denoted by Z

Closure property :

Let a , b ∈ Z

Then sum of two integer is also an integer

Thus a + b ∈ Z

Associative property :

Let a , b , c ∈ Z

Then a + ( b + c ) = ( a + b ) + c

Thus associative property holds

Existence of identity element :

Let a ∈ Z

Then a + 0 = 0 + a = a

So 0 is the additive identity

Existence of inverse :

Let a ∈ Z

Then - a ∈ Z such that

a + ( - a ) = ( - a ) + a = 0

So - a is the additive inverse of a

Hence ( Z , + ) is a group

Abelian property :

Let a , b ∈ Z

Then a + b = b + a

Thus abelian property holds

Hence ( Z , + ) is an abelian group

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