is the set of all positive integers closed under addition?

if yes..give 2 examples

if no,,,give 2 examples that prove it is not

adding positive integers results in a positive integer

... this is the definition of a closed set

pick any examples that you like

The set of all positive integers is closed under addition. To understand why, we need to recognize that closure under addition means that if you add any two positive integers together, the result will always be another positive integer.

To prove that the set of positive integers is closed under addition, let's take two positive integers:

Example 1:
Let's choose the numbers 5 and 3.
When we add 5 and 3 together, we get 8, which is also a positive integer.

Example 2:
Let's choose the numbers 10 and 7.
When we add 10 and 7 together, we get 17, which is also a positive integer.

In both examples, we can observe that by adding any two positive integers, we always get another positive integer. Hence, the set of positive integers is closed under addition.

However, if the question were asking about the set of natural numbers (including 0), it would not be closed under addition. Here are two examples to demonstrate this:

Example 1:
Let's choose the numbers 2 and 3.
When we add 2 and 3 together, we get 5, which is a positive integer, but not a natural number since it does not include zero.

Example 2:
Let's choose the numbers 0 and 1.
When we add 0 and 1 together, we get 1, which is a natural number, but in this case, it's not positive.

Therefore, the set of natural numbers (including zero) is not closed under addition.