Question 3

Natural numbers are not closed under subtraction.

Step 1 — Understanding Closure

We need to define closure. Closure means the result stays in the set. Let's take two numbers from a set. We perform an operation on them. The result must also be in that set.

Step 2 — Testing Subtraction Example 1

Let's test subtraction with an example. Consider the natural numbers 5 and 3. We subtract 3 from 5. $5 - 3$ $= 2$

$\boxed{2 \text{ is a natural number}}$ This example shows a natural number result.

Step 3 — Testing Subtraction Example 2

Let's try another example. Consider the natural numbers 3 and 5. We subtract 5 from 3. $3 - 5$ $= -2$

$\boxed{-2 \text{ is not a natural number}}$ A natural number cannot be negative. Since we found a case where the result is not a natural number, the set is not closed.

Answer

(i) Natural numbers are not closed under subtraction. (ii) For example, $5 - 3 = 2$. Here, 2 is a natural number. (iii) However, $3 - 5 = -2$. Here, -2 is not a natural number.