Let $\displaystyle V = \left\{x,y}\right\}$ be a set with exactly two vectors, x and y. Define vector addition and scalar multiplication in V by the following rules:

Vector addition: $\displaystyle x+x=x$, $\displaystyle y+y=x$, $\displaystyle x+y=y$, and $\displaystyle y+x=y$.

Vector multiplication: $\displaystyle cx=x$, and $\displaystyle cy=y$ for all $\displaystyle c \in \mathbb{R}$.

Prove that V is not a vector space by finding one axiom in the definition of a vector space that fails to hold. You must state the axiom clearly and show it does not hold.

I'm at a loss on this one.