Integers are closed:
If y is an integer, so is its absolute value |y|. So, x + |y| is a sum between two integers (it's either x + y or x - y, depending on the sign of y), and the sum between two integers is an integer.
Negative integers are not closed:
If you pick x and y such that [tex]0>x>y[/tex], you'll get a counterexample.
For example, if we choose [tex]x=-1,\ y=-10[/tex] we have
[tex]x * y = -1 + |-10| = -1+10 = 9[/tex]
which is positive. So, we started with two negative numbers and we got a positive number, which means that negative numbers are not closed under this operator.
Positive integers are closed:
If x and y are both positive, then the absolute value is useless, because |y| = y. So, we have
[tex] x * y = x + |y| = x+y[/tex]
And our operator * is the usual sum. And the sum of two positive numbers is a positive number, so positive numbers are closed.
Multiples of 3 are closed.
If x and y are multiple of 3, then there exist integers m and n such that
[tex]x=3n,\ y=3m[/tex]
This implies that
[tex]x*y = 3n + |3m| = 3n+3|m| = 3(n+|m|)[/tex]
which is a multiple of 3, because we wrote it as "3 times something".
So, if we start with two multiples of 3 we get another multiple of 3, which means that multiples of 3 are closed.
