The World of Numbers | Exercise 3.1

Question 3

We know that Natural Numbers are closed under addition (the sum of any two natural numbers is always a natural number). Are they closed under subtraction? Provide a couple of examples to justify your answer.

Solution

Understand the Question

A set is closed under an operation if performing that operation on any two elements of the set always produces an element within the same set.
The set of natural numbers is $\mathbb{N} = \{1, 2, 3, 4, \dots\}$ .
To determine if natural numbers are closed under subtraction, we check whether subtracting any two natural numbers always yields a natural number.

Step 1 · Test Subtraction with Examples

Consider the natural numbers $5$ and $3$ : $5 - 3 = 2$ Here, $2$ is a natural number.

Now, reverse the order and subtract $5$ from $3$ : $3 - 5 = -2$ Here, $-2$ is a negative integer and is not a natural number.

Since the result is not always a natural number, natural numbers are not closed under subtraction.

Answer

No, natural numbers are not closed under subtraction (e.g., $5 - 3 = 2 \in \mathbb{N}$ , but $3 - 5 = -2 \notin \mathbb{N}$ ).

Common Mistakes

Generalizing from a Single Positive Case: Assuming natural numbers are closed under subtraction just because $5 - 3 = 2$ is a natural number. Closure requires the property to hold for every pair of numbers.
Confusing Natural Numbers with Integers: Assuming negative numbers like $-2$ are natural numbers. Natural numbers only include positive counting numbers $\{1, 2, 3, \dots\}$ .

Question 3

More questions in Exercise 3.1