Answer to Question #120558 in Discrete Mathematics for inv

Question

Justify if the following operations on relevant sets are binary operations or not.
1) Multiplication and Division on se of Natural numbers
2) Subtraction and Addition on Set of Natural numbers
3) Exponential operation: (x,y)→x^y on Set of Natural numbers and set of Integers

Answer 1

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

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

The FUNCTION O:AXA→A is defined as o(a,b)=aob is called binary operation if aob∈ A

EXAMPLE : ADDITION

+:NXN → N,

+(a,b)=a+b

a,b∈ 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

EXAMPLE : subtraction

-:NXN → N,

-(a,b)=a-b

a,b∈ N

2-=-1

2,3 belongs to natural number and resultant(-1 is not 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)