A set of integers is closed under division if, for any two integers a and b (where b is not zero) in the set, the result of a ÷ b is also an integer within that set.
Understanding Closure Under Division
The concept of closure in mathematics means that performing an operation (in this case, division) on elements within a set always results in another element that is also within that set. For integers and division, this is rarely true.
Why Integers Are Generally Not Closed Under Division
Most divisions involving integers will result in a rational number (a fraction) that is not an integer. For instance, 5 ÷ 2 = 2.5, which is not an integer. Therefore, the set of all integers is not closed under division.
Examples of Sets of Integers That Could Be Closed Under Division
Finding sets of integers closed under division is difficult, and they tend to be very limited:
- The set {1, -1}: 1 ÷ 1 = 1, 1 ÷ -1 = -1, -1 ÷ 1 = -1, -1 ÷ -1 = 1. All results are within the set.
- The set {0, 1, -1}: Requires careful handling of division by zero, which is typically undefined. However, if we exclude division by zero, we still fail because 0 ÷ 1 = 0, 0 ÷ -1 = 0; but if we consider 1 ÷ 1= 1, 1 ÷ -1 = -1, -1 ÷ 1 = -1, and -1 ÷ -1 = 1, these results are within the set.
Important Considerations
- Division by Zero: Division by zero is undefined in standard mathematics. Any set claiming closure under division must specifically address or exclude zero in the denominator.
Conclusion
The set of all integers is not closed under division. Specific, limited sets of integers, like {1, -1}, can be closed under division, but they are exceptions rather than the rule. The crucial aspect is that the result of dividing any two elements within the set (excluding division by zero) must also be a member of that same set.
