Problem understanding the definition of Quotient spaces

Question

Quotient spaces are defined as $V/U := \{a + U | a \in V\} $ where $a + U := \{a + u | u \in U\}$ U is a subspace of V, so $u \in U \implies u \in V$ and $\{a + u | a \in V\}$ = V. How does this not mean that $V/U = V$ according to the definition?

Answer 1

Maybe it would help to think about the example where $V = \mathbb R^2$ and $U$ is the subspace $\{ (x,y) \mid y = x \}$. If $a \in \mathbb R^2$, then $a + U$ is a line parallel to the line $y = x$. So each element of $V/U$ is a line in $\mathbb R^2$ with slope $1$.