Question

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

Answer 1

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.