Is natural numbers closed under subtraction?

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.

Natural numbers are an important part of the number system, including all the positive integers from 1 to infinity, used for counting purposes. Natural numbers come under real numbers and include the positive integers 1, 2, 3, 4, 5, 6, 7, 8... and so on.

Numbers can be found everywhere around us, used for counting objects, representing or transferring money, calculating temperature, telling time, and so on. "Natural Numbers" refer to the Numbers used to count objects. When counting objects, we might say 5 glasses, 6 books, 1 bottle, and so on.

The number system includes all positive integers from 1 to infinity, which is known as Natural Numbers. Natural Numbers are sometimes known as counting numbers because they do not include zero or negative numbers. They are only positive integers, not zeros, fractions, decimals, or negative Numbers, and they are part of the real Number system.

A set of all whole numbers except 0 is referred to as Natural Numbers. These figures play a significant role in our day-to-day activities and communication.

Natural Numbers are those that can be counted and are a portion of real Numbers. The set of Natural Numbers contains only positive integers such as 1, 2, 3, 4, 5, 6, and so on.

Natural Numbers refer to non-negative integers (all positive integers). Examples can be 39, 696, 63, 05110, and so on.

Natural numbers are the positive integers, including numbers from 1 to infinity. Natural numbers are countable numbers and are preferable for calculations. 1 is the smallest natural number and the sum of natural numbers from 1 to 100 is n(n+1)2.

Whole Numbers and Natural Numbers

Natural numbers and whole numbers are different from each other in the matter of including zero. Whole numbers include zero, but all natural numbers are the positive numbers excluding zero. 

Every natural number is a whole number, but every whole number is not a natural number.

Set of Natural Numbers

The term "Set" refers to a group of items (Numbers in this context). In mathematics, the Set of Natural Numbers is written as 1,2,3,... The Set of Natural Numbers is symbolised by the symbol N. N = 1,2,3,4,5 and so on. In mathematics, the Set of Natural Numbers is written as 1,2,3,...

N is the natural numbers’ set representation and represents the following:

Statement:

N = Set of numbers starting from 1 and lasting till infinity.

Roster Form:

N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10... and so on}

Set Builder Form:

N = {x: x is a number starting from 1}

Properties of the Natural Number

Natural numbers follow four main properties, which are as follows:

1. Closure Property

2. Commutative Property

3. Associative Property

4. Distributive Property

Closure Property

A natural number is closed under addition and multiplication. This means that adding or multiplying two natural numbers results in a natural number. However, for subtraction and division, natural numbers do not follow closure property.

Addition

When a and b are two natural numbers, a+b is also a natural number. For example, 2+3=5, 6+7=13, and similarly, all the resultants are natural numbers.

Subtraction

For two natural numbers a and b, a-b might not result in a natural number. E.g. 6-5 = 1 but 5-6=-1.

Multiplication

When a and b are two natural numbers, a*b is also a natural number. Example, 3*5 =15, and similarly all resultants from multiplication are natural numbers.

Division

For the two rational numbers a and b, the division might or might not result in a natural number. E.g. \[\frac{10}{2} =5\]  but \[\frac{10}{3} = 3.33.\].

Associative Property

Natural numbers follow associative property for addition and multiplication. For three rational numbers, say, a, b and c, a + (b + c) = (a + b) + c and a * (b * c) = (a * b) * c. Whereas, natural numbers do not follow associative property for multiplication and division.

Addition

For natural numbers a, b and c, addition is associative, i.e. a + (b + c) = (a + b) + c. For example, (15 +3) +1 = 19 = 15 + (3 + 1)

Multiplication

For natural numbers a, b and c, multiplication is associative, which means, a * (b * c) = (a * b) * c. Example: (3 * 1) * 15 = 45 = 3 * (1 * 15).

Subtraction 

For three natural numbers a, b, and c, subtraction is not associative, meaning, a – (b – c) is not equal to (a –b) – c. For example: (2 – 15) – 1 = -14 but 2 – (15 – 1) = -12.

Division

For three natural numbers a, b, and c, division is not associative, i.e. \[\frac{a}{(b/c)}\] is not equal to \[\frac{(a/b)}{c}\] . Example: \[\frac{2}{(3/6)} = 4\] but \[\frac{(2/3)}{6} = 0.11\] 

Commutative Property

For any two given natural numbers a and b, addition and multiplication are commutative, i.e. a+b = b+a and a*b = b*a. However, division and subtraction are not commutative for the natural number (s), i.e. a-b is not equal to b-a and \[\frac{a}{b}\]  is not similar to \[\frac{b}{a}\].

Distributive Property

For the given three natural numbers a, b and c, multiplication is distributive over addition and subtraction. This means that a * (b + c) = ab + ac and a * (b – c) = ab – ac.

Smallest Natural Number

1 is the Smallest Natural Number. We know that the Smallest element in N is 1 and that for each element in N, we may talk about the next element in terms of 1 and N. (which is 1 more than that element). 2 is one greater than one, 3 is one greater than two, and so on.

Which is closed under subtraction?

Which is not closed under subtraction?

Is whole numbers closed under subtraction?

Can you subtract natural numbers?