Trying to find the error in my attempt at basic probability set complement problem I am working through a textbook on probability for actuaries and I am having trouble with this problem:

In a universe $U$ of $100$, let $A$ and $B$ be subsets of $U$ such that $|A \cup B| = 70$ and $|A \cup B'| = 90$. Then what is $|A|$?

So here is my thinking: I know $|B|+|B'|=100$, so I would think that I can just subtract $100$ from the sum $|A \cup B| + |A \cup B'| = 160$. Then we have $|A| + |A| = 60$, so $|A| = 30$.
However, the right answer seems to be $60$. Was my method close, or is it just a coincidence that I arrived at exactly half that number?

Use the additivity for cardinality of the union of disjoint sets.

${\def\abs#1{{\lvert #1 \rvert}}\abs {S\cup T}=\abs S+\abs T}$ when $S,T$ are disjoint (finite) sets.

$${\def\abs#1{\lvert #1\rvert}\begin{split} \abs {A\cup B}+\abs{A\cup B'} &=\abs {A\cup(A^\complement\cap B)}+\abs {A\cup(A^\complement\cap B^\complement)}\\ &=\abs A+\abs{A^\complement\cap B}+\abs A+\abs{A^\complement\cap B^\complement}\\ &=2\abs A+\abs {A^\complement\cap(B\cup B^\complement)}\\ &=\abs A+\abs {A\cup A^\complement}\\[2ex] \abs A&=\abs{A\cup B}+\abs{A\cup B^\complement}-\abs U\end{split}}$$
