Question

2. Justify whether the given operations on relevant sets are binary operations or not.

i. Multiplication and Division on set of Natural numbers

ii. Subtraction and Addition on Set of Natural numbers

iii. Exponential operation: 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 belong to the set of natural numbers and the result 5 also belongs to the set natural numbers so addition is a binary operation.

EXAMPLE : subtraction

-:NXN → N,

-(a,b)=a-b

a,b∈ N

2-3=-1

2, 3 belong to the set of natural numbers and the result (-1) does not belong to the set of natural numbers so subtraction is not a binary operation.

EXAMPLE : multiplication

*:NXN→ N,

*(a,b)=a*b

a,b∈ N

2*3=6

2, 3 belong to the set of natural numbers and the result (6) also belongs to the set of natural numbers so multiplication is a binary operation.

EXAMPLE : division

:NXN→ N,

%(a,b)=a%b

a,b∈ N

2%3=0.6

2, 3 belong to the set of natural numbers and the result (0.6) does not belong to the set of natural numbers so division is not a binary operation.

EXAMPLE : exponential

^:NXN→ N,

^(a,b)=a^b

a,b∈ N

2³=8

2, 3 belong to the set of natural numbers and the result (8) also belongs to the set of natural numbers 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 belong to the set of integer numbers and the result (1%8) does NOT belong to the set of integer numbers so exponential is not a binary operation.