I'm stuck at a problem related to vector space detection. I know how to solve it but don't understand a particular information.

We know that if a set of objects follow 10 axioms of vector space than that defines a vector space.

Say we have a problem:

The set of all positive real numbers with operations

And asked to verify if the set is vector space or not.

Now one of the axioms to be satisfied is:

For each

in

, there is an object

in

, called a negative of

, such that

For the above problem an object

then

So,

Now my question: why's

How's that possible? In what way

?

Shouldn't we say that this set is not a vector space because it violates one of the axioms?

I can't seem to find the answer. Can anyone help me on this?