[Math] Proof these operations is not a vector space

Question

Let $\mathbb{R}$ be the set of all real numbers. Define scalar multiplication by

$\alpha x = \alpha \cdot x$ (the usual multiplication of real numbers)

and define addition by

$x \oplus y = \max(x, y)$ (the maximum of two numbers)

Prove that $\mathbb{R}$ with these operations is NOT a vector space. Which of the eight axioms fail to hold?

Please help I know I need to prove it by using the 8 axioms but I get confused with the addition part.

Answer 1

I believe your addition does not have inverse element, so the following is not true: $$\forall v \in V, \exists w(\text{also called }-v) : v \oplus w = 0$$ because $max(v,w)$ can be only $v$ or $w$ but not $0$.