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Which operations is the following set closed under? Explain your answer.

{2,4,6,…}

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Answer: The set {2,4,6,…} is the set of all positive even integers. A set is closed under an operation if applying the operation to any two elements of the set produces another element of the same set. For example, the set {2,4,6,…} is closed under addition, because adding any two positive even integers results in another positive even integer. However, the set is not closed under subtraction, because subtracting a larger positive even integer from a smaller one results in a negative integer, which is not in the set.

The set {2,4,6,…} is closed under the following operations:

Addition: For any positive even integers x and y, x + y is also a positive even integer. For example, 2 + 4 = 6, 6 + 8 = 14, etc.

Multiplication: For any positive even integers x and y, x * y is also a positive even integer. For example, 2 * 4 = 8, 6 * 8 = 48, etc.

Division by 2: For any positive even integer x, x / 2 is also a positive even integer. For example, 2 / 2 = 1, 4 / 2 = 2, etc.

The set {2,4,6,…} is not closed under the following operations:

Subtraction: As mentioned above, subtracting a larger positive even integer from a smaller one results in a negative integer, which is not in the set. For example, 2 - 4 = -2, 6 - 8 = -2, etc.

Division by any other positive integer: For any positive even integer x and any other positive integer y (except 2), x / y may not be a positive even integer. For example, 4 / 3 = 1.333…, 6 / 5 = 1.2, etc.

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