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## Here's the Solution to this Question

1.Properties of Binary operation on a set A are as follows:

a. Closure Property: Consider a binary operation * on A. Then A is closed under the operation *, which means $a * b ∈ A$ , where a and b are elements of A.

b. Associative Property: Consider a binary operation * on A. Then the operation * on A is associative, if for every a, b, c, ∈ A, we have $(a * b) * c = a* (b*c).$

c. Commutative Property: Consider a binary operation * on A. Then the operation * on A is associative, if for every a, b, ∈ A, we have $a * b = b * a.$

d. Identity: Consider a binary operation * on A. Then the operation * has an identity property if there exists an element e in A such that $a * e (right identity) = e * a (left identity) = a ∀ a ∈ A.$

e. Inverse: Consider 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.

f. Distributivity: Consider 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]

2. Let N={1,2,3,...} and Z={...-2,-1,0,1,2,...} be the sets of natural numbers and integers respectively.

i. Multiplication operation on the set N is a binary operation as it follows all the aforementioned properties. However, Division operation on the set N is not a binary operation. It is because it does not satisfy the closure property. Since, division of 2 natural numbers does not always give a natural number.

$eg. 2/4=1/2=0.5$ and 0.5 is not a natural number.

ii. Addition operation on the set N is a binary operation as it follows all the aforementioned properties. However, Subtraction operation on the set N is not a binary operation. It is because it does not satisfy the closure property. Since, subtraction of 2 natural numbers does not always give a natural number.

$eg: 2-3=-1$ and -1 is not a natural number.

iii. Exponential operation on the set N is a binary operation as it follows all the aforementioned properties. However, Exponential operation on the set Z is not a binary operation. It is because it does not satisfy the closure property. Since, exponentiation of two integers may not always yield an integer.

$eg. 2^{-1} =1/2=0.5$ , and 0.5 is not an integer.