Closure property under subtraction in a set ensures that: a) Subtracting any two elements results in a positive number b) Subtracting any two elements results in a number within the set c) Subtracting any two elements results in a negative number d) Subtracting any two elements results in zero
The Correct Answer and Explanation is:
Correct answer: b) Subtracting any two elements results in a number within the set
Explanation:
The closure property refers to whether an operation performed on any two elements of a set always produces a result that also belongs to that set. When discussing the closure property under subtraction, the focus is on whether subtracting any two elements from a given set always results in a number that is also in that same set.
For a set to be closed under subtraction, the difference between any two elements taken from the set must still belong to the set. This does not require the result to be positive, negative, or zero. It only requires the result to remain inside the set.
Let us consider the following example using the set of integers, which is closed under subtraction:
- 7 and 3 are both integers.
- 7 − 3 = 4, which is also an integer.
- 3 − 7 = −4, which is still an integer.
No matter which two integers you choose, their difference is always another integer. Hence, integers are closed under subtraction.
Now let us consider the set of natural numbers, which is not closed under subtraction:
- 5 and 2 are natural numbers, and 5 − 2 = 3, which is fine.
- But 2 − 5 = −3, which is not a natural number, so the closure fails.
Therefore, the key idea is not whether the result is positive, negative, or zero, but whether the result remains in the original set. This is exactly what option b states.
To summarize, option b is correct because it directly describes what closure under subtraction means: the result of subtracting any two elements stays within the set.
