Proof of two properties of orthogonal complements Assume that $X_1\subseteq V$ and $X_2 \subseteq V$
How can we prove that:
1) If $X_1 \subseteq X_2$ then $X_2^\perp\subseteq X_1 ^\perp$
2)  a) $(X_1+X_2)^\perp=X_1^\perp\cap X_2^\perp$ and  b) $(X_1\cap X_2)^\perp=X_1^\perp+X_2^\perp$
I found these 2 posts which are related to at least one of these problems, I could not comprehend the proof in the first post and the second post was unanswered. I was also hinted that I should use the definition of the orthogonal supplement to prove at least the first one but I couldn't do it despite my efforts.
orthogonal complement of a sum
Two proof problems about orthogonal complement
Edit: This is the proof I came up for (2a) after the help I got from the answers:
$\begin
{align}X_1^\perp\cap X_2^\perp=\{y:\langle y,x\rangle=0\ \ \forall x_1\in X_1,\ \ \forall x_2\in X_2\}=\{y\in V:\langle x_1,y\rangle=0\ \ \& \ \ \langle x_2,y\rangle=0\ \ \forall x_1\in X_1,\ \ \forall x_2\in X_2\}=\{y\in V:\langle x_1,y\rangle + \langle x_2,y\rangle=0 \ \ \forall x_1\in X_1, \ \ \forall x_2\in X_2 \}=(X_1+X_2)^\perp
\end{align}$
Is this correct? 
As for (2b), I think it's $X_1^\perp + X_2^\perp = \{y_1 + y_2 : y_1 \in X_1^\perp \text{ and }y_2 \in X_2^\perp\}\\$  we got to work with, but its definition seems to differ a bit compared to the other ones on the list. I'm not sure how I'm supposed to proceed with it to reach the desired result, which would be: 
$(X_1 \cap X_2)^\perp = \{y : \langle y,x \rangle = 0 \text{ for all } x \in X_1 \cap X_2\}$
 A: Here's an answer to the first part. 
We want to show that if $y \in X_2^\perp$, then $y \in X_1^\perp$. Consider any $y \in X_2^\perp$. By definition, this means that we have $\langle y,x \rangle = 0$ for every $x \in X_2$.  
Now, consider any $x \in X_1$. It is also true that $x \in X_2$. So from what we just said, it follows that $\langle y, x \rangle = 0$. 
Because $\langle y,x \rangle = 0$ for every $x \in X_1$, $y$ is an element of $X_1^\perp$, which is what we wanted.
Answering the second part requires that you similarly "unpack" the definition of the orthogonal complement.

In response to your comments: here is a way to write out the definition of all relevant sets:
$$
\begin{align}
X_1 + X_2 &= \{x: x = x_1 + x_2 \text{ with } x_1 \in X_1,x_2 \in X_2\}
\\ & = \{x_1 + x_2 : x_1 \in X_1, x_2 \in X_2\}\\
(X_1 + X_2)^\perp &= \{y: \langle y,x \rangle = 0 \text{ for all } x \in X_1 + X_2\}\\
&= \{y: \langle y,x_1 + x_2 \rangle = 0 \text{ for all } x_1 \in X_1, x_2 \in X_2\}\\
X_1^\perp + X_2^\perp &= \{y_1 + y_2 : y_1 \in X_1^\perp \text{ and }y_2 \in X_2^\perp\}\\
X_1 \cap X_2 &= \{x: x \in X_1 \text{ and }x \in X_2\}\\
X_1^\perp \cap X_2^\perp &= \{y: y \in X_1^\perp \text{ and } y \in X_2^\perp\}
\\ &= 
\{y : \langle y,x \rangle = 0 \text{ for all } x \text{ such that } x \in X_1 \text{ and all } x \text{ such that } x \in X_2\}\\
(X_1 \cap X_2)^\perp &= \{y : \langle y,x \rangle = 0 \text{ for all } x \in X_1 \cap X_2\}
\end{align}
$$
