Question

The set of natural number is closed under the binary operations of:
A) Addition, subtraction, multiplication and division.
B) Addition, subtraction, multiplication but not division.
C) Addition and multiplication but not subtraction and division.
D) Addition and subtraction but not multiplication and division.

Answer 1

We will use the definition that a set is closed under any operation if and only if the binary operation on any two elements belonging in the set results in a third number which also belongs to the given set. Otherwise, not closed under the operation. We will check individually for each binary operation given if they satisfy the condition or not.

Complete step by step solution:
Here, a set of natural numbers \mathbb{N} = \left\\{ {1,2,3,4,....} \right\\} is given and we are required to check if $\mathbb{N}$ is closed under the binary operations such as addition, subtraction, multiplication and division.
Let a binary operation be $*$ defined on $\mathbb{N} \to \mathbb{N}$ such that $a * b \in \mathbb{N}$ for $\mathbb{N}$ to be closed under $*$.
We will check individually for all the given operations in the options if the set of natural number is closed under them or not.
Addition ‘$+$’
For the set of natural numbers, addition of two natural numbers will always give another natural number i.e., $a + b \in \mathbb{N}$, $\forall a,b \in \mathbb{N}$. Hence, the set of natural numbers is closed under addition.
Subtraction ‘$-$’
For the set of natural numbers, subtraction of two numbers may or may not produce a natural number i.e., for $5 \in \mathbb{N},9 \in \mathbb{N}$, $5 - 9 = - 4 \notin \mathbb{N}$. Hence, the set of natural numbers is not closed under subtraction.
Multiplication ‘$\times$’
For the set of natural numbers, multiplication of any two numbers always results in a natural number i.e., $a \times b \in \mathbb{N}$, $\forall a,b \in \mathbb{N}$. Hence, the set of natural numbers is closed under multiplication.
Division ‘$\div$’
For the set of natural numbers, division of two numbers may not necessarily produce a natural number i.e., for $13 \in \mathbb{N},7 \in \mathbb{N}$, $13 \div 7 = \dfrac{{13}}{7} \notin \mathbb{N}$. Hence, the set of natural numbers is not closed under division.
Hence, we can say that the set of natural numbers is closed under addition and multiplication but not under subtraction and division.

Therefore, option (C) is correct.

Note:
In this question, you may get the wrong idea of any set being closed under any binary operation. It simply means that if a set is closed under any binary operation, then all the elements obtained after applying the operation must lie in the set. Even if an element lies outside the set, the set is not closed under that operation. And hence, we can understand why the set $\mathbb{N}$ under operations ‘$-$’ and ‘$\div$’ is considered not closed even though a few elements lie inside the set.