my question asks the following:
Prove the following are members of the set of all positive real numbers
$$x+y=xy$$
$$kx=x^k$$
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Hmm. What is the question again? That the set of all positive numbers have those properties???? Clearly false! My educated guess is that the question really is to prove that the set of all positive real numbers forms a vector space if we redefine the "sum" of vectors $x$ and $y$ to be $xy$ and the "scalar multiple" of the vector $x$ by scalar $k$ (so $k$ is not restricted to being positive) is $x^k$. Please check! It looks like you haven't really understood what you need to show (that's ok - many newbies have trouble getting used to what axioms really mean). – Jyrki Lahtonen Sep 14 '16 at 04:37
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How to prove this is a vectorspace?
Show all of the axioms are true.
The "hard" part is identifying the zero vector for this vector space, and using your proposed zero vector in showing all axioms are satisfied.
To get you started, let $x\in V$. What vector $z$ satisfies $x+z=x$ for all $x\in V$?
David P
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