Justify whether the given operations on relevant sets are binary operations or not. - Multiplication and Division on set of natural numbers - Subtraction and Addition on Set of natural numbers - Exponential operation: (x, y) → xy on Set of Natural numbers and set of Integers.

Solution:

1.

Basically, binary operations on a set are calculations that combine two elements of the set (called operands) to produce another element of the same set.

The binary operation, (say *) on a non-empty set A are functions from A × A to A.

$*: A × A → A$

It is an operation of two elements of the set whose domains and co-domain are in the same set.

Closure property: An operation * on a non-empty set A has closure property, if,

$a ∈ A, b ∈ A ⇒ a * b ∈ A$

2.

i)

Multiplication of two natural numbers is always a natural number.

Division of two natural numbers may not always be a natural number, it may result in fractions(rational numbers). e.g.: $2 \in N, 3 \in N, but (2/3)\notin N$

Thus, Multiplication is a binary operation on the set of Natural numbers whereas Division is not a binary operation on the set.

ii)

Addition of two natural numbers is always a natural number.

Subtraction of two natural numbers does not need always to be a natural number, the result can be negative(integer). e.g.: $2 \in N, 3 \in N, but (2-3)=-1\notin N$

iii)

Exponential operation on two natural numbers always results in a natural number.

However, exponential operation on two integers does not always result in an integer.

eg: $2,3 \in Z; 2^3=8\in Z$. But $2, -3 \in Z; 2^{-3}=1/8=0.125 \notin Z$

Thus, exponentiation is a binary operation on the set of Natural numbers whereas it is not a binary operation on the set of Integers.