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NATURAL NUMBER {1,2,3....}

INTEGER NUMBER {...-3,-2,-1,0,1,2,3....}

Multiplication and Division on set of Natural numbers

The FUNCTION O:AXA$\to$A is defined as o(a,b)=aob is called binary operation if aob$\isin$ A

+:NXN $\to$ N,

+(a,b)=a+b

a,b$\isin$ N

2+3=5

3,2 belongs to natural number and resultant 5 is also belong to natural number so addition is a binary operation

-:NXN $\to$ N,

-(a,b)=a-b

a,b∈ N

2-3=-1

2,3 belongs to natural number and resultant(-1 is not also belong to natural number so subtraction is not a binary operation

EXAMPLE : multiplication

*:NXN→ N,

*(a,b)=a*b

a,b∈ N

2*3=6

2,3 belongs to natural number and resultant(6 is also belong to natural number so multiplication is a binary operation)

EXAMPLE : division

:NXN→ N,

%(a,b)=a%b

a,b∈ N

2%3=0.6

2,3 belongs to natural number and resultant(0.6 is not belong to natural number so division is not a binary operation)

EXAMPLE : exponential

^:NXN→ N,

^(a,b)=a^b

a,b∈ N

23=8

2,3 belongs to natural number and resultant(8 is also belong to natural number so exponential is also a binary operation)

EXAMPLE : exponential

^:IXI→ I,

^(a,b)=a^b

a,b∈ I

2-3=1%8=0.12

2,-3 belongs to integer number and resultant(1%8 is NOT belong to integer number so exponential is not a binary operation)

properties of binary of binary operation

1)commutative

2)associative

3)distributive

4)identity

5)inverse

1)commutative: aob=boa

example 2*3=3*2

6=6

2,3 ,6 belongs to natural number

5=5

2,3 ,5 belongs to natural number

2)associative law:

ao(boc)=(aob)oc

2+(3+2)=(2+3)+2

7=7

3)distibutative:: Consider a non-empty set A, and a binary operation * on A. Then the operation * distributes over +, if for every a, b, c ∈A, we have

a * (b + c) = (a * b) + (a * c)    [left distributivity]

(b + c) * a = (b * a) + (c * a)    [right distributivity]

4)Identity: Consider a non-empty set A, and a binary operation * on A. Then the operation * has an identity property if there exists an element e in A such that a * e (right identity) = e * a (left identity) = a ∀ a ∈ A.

5)Inverse: Consider a non-empty set A, and a binary operation * on A. Then the operation is the inverse property, if for each a ∈A,,there exists an element b in A such that a * b (right inverse) = b * a (left inverse) = e, where b is called an inverse of a.

example aob=boa=e

2+(-2)=2+(-2)=0

2*(1%2)=2*(1%2)=1