A vector space over $\mathbb{R}$ is a set $V$ of objects (called vectors) together with two operations, addition and multiplication by scalars (real numbers), that satisfy the following $10$ axioms. The axioms must hold for all vectors $\mathbf{u}, \mathbf{v}, \mathbf{w}$ in $V$ and for all scalars $\alpha, \beta$ in $\mathbb{R}$.

1. (Closed under addition:) The sum of $\mathbf{u}$ and $\mathbf{v}$, denoted $\mathbf{u} + \mathbf{v}$, is in $V$.
   
   
2. (Closed under scalar multiplication:) The scalar multiple of $\mathbf{u}$ by $\alpha$, denoted $\alpha \mathbf{u}$, is in $V$.
   
   
3. (Addition is commutative:) $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$.
   
   
4. (Addition is associative:) $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$.
   
   
5. (A zero vector exists:) There exists a vector $\mathbf{0}$ in $V$ such that $\mathbf{u} + \mathbf{0} = \mathbf{u}$.
   
   
6. (Additive inverses exist:) For each $\mathbf{u}$ in $V$, there exists a $\mathbf{v}$ in $V$ such that $\mathbf{u} + \mathbf{v} = \mathbf{0}$. (We write $\mathbf{v} = -\mathbf{u}$.)
   
   
7. (Scaling by $1$ is the identity:) $1 \mathbf{u} = \mathbf{u}$.
   
   
8. (Scalar multiplication is associative): $\alpha (\beta \mathbf{u}) = (\alpha \beta) \mathbf{u}$.
   
   
9. (Scalar multiplication distributes over vector addition:) $\alpha (\mathbf{u} + \mathbf{v}) = \alpha \mathbf{u} + \alpha \mathbf{v}$.
   
   
10. (Scalar addition is distributive:) $(\alpha + \beta) \mathbf{u} = \alpha \mathbf{u} + \beta \mathbf{u}$.
    
    

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Let $V$ be the set $\lbrace (x,y) \mid x, y \in \mathbb{R} \rbrace$ with addition operation $\oplus$ defined by $(x_1,y_1) \oplus (x_2,y_2) = (x_1+x_2, y_1 y_2)$ and scalar multiplication $\odot$ defined by $\alpha \odot (x,y) = (x,0)$.

Show that $V$ is not a vector space by determining which of the $10$ vector space axioms are not true for $V$. Enter your answer as a comma separated list such as 3, 5, 6 if axioms numbered 3, 5 and 6 fail to be true.


In the box below, explain why each vector space axiom you chose in the previous part fails to be true.