Tessellate S.T.E.M.S - Mathematics - College - Set 2 - Problem 1

Consider $X=[0,1)$ .

For $a, b \in X$ , we define an operation $\oplus$ by $a\oplus b= \{a+b\}$ .

For $a \in \mathbb{R}, b \in X$ , we define an operation $\otimes$ by $a\otimes b= \{ab\}$ .

$\textbf{Statement 1.}$ $X$ is a $\mathbb{Q}$ -vector space with addition $\oplus$ and $\mathbb{Q}$ -multiplication $\otimes$

$\textbf{Statement 2.}$ $X$ is a $\mathbb{R}$ -vector space with addition $\oplus$ and $\mathbb{R}$ -multiplication $\otimes$

Which of the above statements are correct?

$\textbf{Note}$ : $\{m\}$ denotes the fractional part of $m$ .

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This problem is a part of Tessellate S.T.E.M.S.