The set of integers demonstrates closure under which operation? a) Addition only b) Multiplication only c) Addition, subtraction, and multiplication d) Addition, subtraction, multiplication, and division
The Correct Answer and Explanation is:
The correct answer is c) Addition, subtraction, and multiplication.
Explanation:
Closure is a property that a set has when performing a particular mathematical operation on any two elements of the set always results in another element from the same set. In the case of integers, the set of integers (denoted as ℤ) includes all the positive and negative whole numbers, including zero.
Let’s analyze the operations one by one:
- Addition:
- If you add two integers, the result is always an integer. For example, 3+(−5)=−23 + (-5) = -23+(−5)=−2, which is an integer. Therefore, integers are closed under addition.
- Subtraction:
- When you subtract two integers, the result is also an integer. For example, 4−7=−34 – 7 = -34−7=−3, which is an integer. Thus, integers are closed under subtraction.
- Multiplication:
- Multiplying two integers always results in an integer. For example, (−2)×6=−12(-2) \times 6 = -12(−2)×6=−12, which is an integer. So, integers are closed under multiplication.
- Division:
- However, division does not always result in an integer. For example, 7÷3≈2.337 \div 3 \approx 2.337÷3≈2.33, which is not an integer. This means that integers are not closed under division.
Since integers are closed under addition, subtraction, and multiplication but not under division, the correct answer is c) Addition, subtraction, and multiplication.
