Evaluating $|A\cup B|$ 
$A$ and $B$ are the subsets of the same universal set
$$|A' \cap B| = 2$$
$$|B'-A| = 4$$
$$|A-B|' = 11$$
$$|B|' = 13$$
Evaluate $|A\cup B|$

There are too many equations so I could not solve this problem using Venn Diagram. Perhaps I have to go more conceptually. Could you assist me? 
Regards
 A: Just use the following two relations for any two sets $M,N$:


*

*$M \setminus N = M \cap N'$

*$|M| = |M \cap N| + |M \cap N'|$
Assuming you mean $|()'|$ instead of $|()|'$  and setting $U$ to be the universe here you may proceed as follows:


*

*$|B' \setminus A| = \color{blue}{|B' \cap A'| = 4}$

*$|A'| =|A' \cap B| +  |A' \cap B'| = 2 + 4 = 6$

*$11 =|(A\setminus B)'| =|(A \cap B')'| = |A' \cup B| =  |A'| + |B \cap A| = 6 + |B \cap A| \Rightarrow |B \cap A | = 5$

*$|B| = |B \cap A |  + |B \cap A' | = 5 + 2 = 7$

*$13 = |B'| = |U| - |B| = |U| - 7 \Rightarrow \color{blue}{|U| = 20}$
Finally
$$\boxed{|A \cup B|} = \color{blue}{|U|} - \color{blue}{|B' \cap A'|} = 20 - 4 \boxed{= 16}$$
A: First note that $|A-B|'$ and $|B|'$ do not make any sense, because $|X|$ turns a set $X$ into a number. And you can't take the complement of a number, only of a set.
So $|A-B|'$ should be $|(A-B)'|$ or $|A-B'|$ and $|B|'$ should be $|B'|$.
If you think it's hard to construct a Venn diagram, you just do it with little steps.
For example with $(A - B)'$: first you draw the Venn diagrams of $A$ and $B$, then $A - B$ and then $(A - B)'$.
$|A' \cap B| = 2$

$|B' - A| = 4$

$|(A - B)'| = 11$ or $|A - B'| = 11$


$|B'| = 13$

$|A \cup B| =\ ?$

Now you do the math!
