1. consider the operation * defined by x* y = x + lyl.
which of the following sets are closed under this operation?
- integers
negative integers
positive integers
integer multiples of 3

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 , you'll get a counterexample.

For example, if we choose we have

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

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

This implies that

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.

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