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.
No, Natural Numbers are not closed under subtraction. Closure under subtraction means that the difference between any two natural numbers must always be a natural number. However, this property does not hold for natural numbers.
When we subtract a larger natural number from a smaller one, the result is a negative number. For example, 3 - 5 = -2. Here, -2 is a negative number (representing debt), which does not belong to the set of natural numbers {1, 2, 3, ...}.
Furthermore, subtracting a natural number from itself yields zero, which is also not a natural number. For example, 5 - 5 = 0. Since the difference of two natural numbers can be a negative number or zero, they are not closed under subtraction.
Explanation
The textbook defines Natural Numbers as the counting numbers {1, 2, 3, ...}. It also explains that subtracting a larger number from a smaller one (like 3 - 5) results in a negative number, and subtracting a number from itself (5 - 5) results in zero. Since neither negative numbers nor zero are part of the set of natural numbers, the set is not closed under subtraction.
Solution Steps
Step 1: Define closure under subtraction (difference of any two numbers in the set must also be in the set).
Step 2: Test by subtracting a larger natural number from a smaller one: 3 - 5 = -2. Since -2 is a negative number and not a natural number, closure fails.
Step 3: Test by subtracting a natural number from itself: 5 - 5 = 0. Since 0 is not a natural number, closure fails again.