Under what operations are the set of integers closed? Explain your answer. UPDATE ANSWER IS -The set of integers is closed under addition, subtraction, and multiplication. -The set of integers is not closed under division. -Give a counterexample to show that the integers are not closed under division. For example, 1 divided by 3 is not an integer.
The Correct Answer and Explanation is:
The set of integers is closed under addition, subtraction, and multiplication. This means that when you perform any of these operations with integers, the result will always be an integer.
- Addition: If you take two integers and add them together, the sum will always be an integer. For example, 3+5=83 + 5 = 83+5=8, where both 3 and 5 are integers, and the sum (8) is also an integer.
- Subtraction: If you subtract one integer from another, the result will also be an integer. For example, 7−4=37 – 4 = 37−4=3, which is still an integer.
- Multiplication: When you multiply two integers, the result will always be an integer. For example, 2×6=122 \times 6 = 122×6=12, where both 2 and 6 are integers, and the product (12) is an integer.
However, division does not always result in an integer. The set of integers is not closed under division. This is because when you divide one integer by another, the result may not be an integer. A counterexample is dividing 1 by 3. Mathematically, 13=0.333…\frac{1}{3} = 0.333…31=0.333…, which is not an integer but a rational number. Therefore, this example demonstrates that integers are not closed under division.
In conclusion, integers are closed under addition, subtraction, and multiplication, but not under division, as division can produce non-integer results.
