Understanding Closure Under Division
Closure under an operation means that when you perform that operation on members of a set, the result is also a member of that set. Let’s analyze whether integers are closed under division.
Sets to Consider
- Integers (Z): The set of all positive and negative whole numbers, including zero.
- Rational Numbers (Q): Numbers that can be expressed as the quotient of two integers, where the denominator is not zero.
- Whole Numbers (W): The set of all non-negative integers (0, 1, 2, …).
- Naturals (N): The set of all positive integers (1, 2, 3, …).
Analyzing Closure Under Division
Let’s see if each of these sets is closed under division:
- Integers: When you divide one integer by another, you generally do not get an integer. For example, dividing 1 by 2 gives 0.5, which is not an integer. Thus, the set of integers is not closed under division.
- Rational Numbers: By definition, every division of two rational numbers yields another rational number. For instance, dividing 2/3 by 4/5 results in (2/3) * (5/4) = 10/12, which simplifies to 5/6, a rational number. Therefore, the set of rational numbers is closed under division.
- Whole Numbers: Similar to integers, dividing whole numbers does not always yield a whole number. For instance, dividing 1 by 2, results in 0.5, which is not a whole number. So the set of whole numbers is not closed under division.
- Naturals: Just like whole numbers, the division of two natural numbers does not always result in a natural number. For example, 1 divided by 2 is 0.5, which is again not a natural number. Therefore, the set of natural numbers is not closed under division.
Conclusion
From our analysis, the only set that is closed under division is the set of rational numbers. The integers, whole numbers, and natural numbers are not closed under this operation as they can yield results that fall outside of their respective sets.