Question

Under which operations is the set of whole numbers {0,1,2,3...} closed

Answer 1

Checking with Addition:

Let x, y be two whole numbers.

Then, x + y is definitely a whole number.

So, the set of whole numbers is closed under addition.

Checking with Subtraction:

Let x, y be two whole numbers.

Then, x - y may or may not be a whole number.

For example, if we take 3 and 2,

3 - 2 = 1 is a whole number but if we take 2 and 5, then

2 - 5 = -3 is not a whole number.

So, the set of whole numbers is not closed under subtraction.

Checking with Multiplication:

Let x, y be two whole numbers.

Then, x × y is definitely a whole number.

So, the set of whole numbers is closed under multiplication.

Checking with Division:

Let x, y be two whole numbers.

Then, x / y may or may not be a whole number.

For example, if we take 6 and 3,

6/3 = 2 is a whole number.

But, if we take 4 and 5,

4/5 is not a whole number.

So, the set of whole numbers is not closed under division.