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.

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). eg: $2 \in N, 3 \in N, but (2/3)\notin N$

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 may not always be a natural number, the result maybe negative(integer). eg: $2 \in N, 3 \in N, but (2-3)=-1\notin N$

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.