Question

Is the set of numbers that can be written as the product of 6 and an integer closed under subtraction?

A.
Yes, because the difference of any two multiples of 6 is equal to 6.

B.
Yes, because the difference of any two multiples of 6 is also a multiple of 6.

C.
No, and a counterexample is 6 – 6 = 0.

D.
No, and a counterexample is 6 – 18 = –12.

Answer 1

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Answer 2

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Answer 3

So, here is my logic. I begin by analyzing the question, they want to know if 6x - 6y is within the domain of 6D. Meaning is the difference between any two multiples of 6 able to be rewritten as 6D where you can make D whatever you want. Even Pizza (just kidding, but you get it.) and since 6x-6y = 6(x-y) which (x-y) can be made into another variable like... Maybe.... D? Then you CAN rewrite 6x-6y = 6(x-y) = 6D where D = (x-y) then it's true! You can write any difference between two multiples of 6 as 6D and so it's closed under Subtraction. When something is closed under something it means when you do that operation (subtraction in this case) it doesn't change the original formula (which in this case was 6x and it doesn't since 6x can equal 6D since we can make D whatever we want).