Part 1 1. Describe the characteristics of different binary operations that are performed on the same set. 2. Justify whether the given operations on relevant sets are binary operations or not. i. Multiplication and Division on se of Natural numbers ii. Subtraction and Addition on Set of Natural numbers iii. Exponential operation: on Set of Natural numbers and set of Integers

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

EXAMPLE : ADDITION

+: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

EXAMPLE : ADDITION

-: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

addition,multiplication is a binary operation and also follow the commutative rule

example 2*3=3*2

6=6

2,3 ,6 belongs to natural number

addition 2+3=3+2

5=5

2,3 ,5 belongs to natural number

2)associative law:

ao(boc)=(aob)oc

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

7=7

addition,multiplication is a binary operation and also follow the associative rule

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

so addition multiplication follow the inverse property