When Does $\operatorname{Span}(A+B)=\operatorname{Span}(A)+\operatorname{Span}(B)$? let there be $A,B \in V$.

$$A+B=\{a+b: a\in A,b \in B\}$$
Does $\operatorname{Span}(A+B)=\operatorname{Span}(A)+\operatorname{Span}(B)$?
I think it is right, even if there are vectors that are both in $A$ and $B$.

To show they are equal, you have to show span{$ A + B $} $\subseteq$ span{$ A$} + span{$B $}, and span{$ A$} + span{$B $} $\subseteq$ span{$ A + B $}.
For span{$ A + B $} $\subseteq$ span{$ A$} + span{$B $}, let $x \in$ span{$ A + B $}.  Then $x = c(a + b)$ for some scalar $c$ and elements $a \in A, b \in B$.  But then that means $x = ca + cb$, and $ca \in$ span{$A$}.  Similarly, $cb \in$ span{$B$}.  So $ca + cb \in $ span{$A$} + span{$B$}.
So we know as we just proved that span{$ A + B $} $\subseteq$ span{$ A$} + span{$B $}.  But the other way does not hold.  Here is a counter example:
Suppose $V = \mathbb{R}^{2}$.  Let $A = \left \{ \begin{bmatrix}1 \\ 0 \end{bmatrix} \right \}$ and $B = \left \{ \begin{bmatrix}0 \\ 1 \end{bmatrix}  \right \}$.  Then span{$A$} $ = \left \{ \begin{bmatrix}c \\ 0 \end{bmatrix} \mid c \in \mathbb{R} \right \}$.  Similarly, span{$B$} $ = \left \{ \begin{bmatrix}0 \\ d \end{bmatrix} \mid d \in \mathbb{R} \right \}$.  Then span{$A$} + span{$B$} $= \left \{ \begin{bmatrix}c \\ d \end{bmatrix} \mid c,d \in \mathbb{R} \right \} = \mathbb{R}^{2}$.  But span{$A + B$} $= \left \{  \begin{bmatrix}c \\ c \end{bmatrix} \mid c \in \mathbb{R} \right \}$.  This clearly is not all of $\mathbb{R}^{2}$, since it only contains vectors where the $x$ and $y$ coordinate are equal.  In particular, $\begin{bmatrix} 2 \\ 3 \end{bmatrix} \in \mathbb{R}^{2} = $ span{$A$} + span{$B$}, but $\begin{bmatrix} 2 \\ 3 \end{bmatrix} \not \in $ span{$A + B$}.
