Exam

Olympiad workbook

Subject

IOQM

Chapter

Sets and Relations

Olympiad workbook · IOQM · Sets and Relations

Let A and B be two finite sets such that there are exactly 144 sets...

Let $A$ and $B$ be two finite sets such that there are exactly 144 sets which are subsets of $A$ or subsets of $B$. Find the number of elements in $A \cup B$.

Step-by-step solution

$$\begin{aligned} & 2^{a+b}+2^{b+c}-2^b=144=2^4 \cdot 9 \\ \Rightarrow & 2^b\left(2^a+2^c-1\right)=2^4 \cdot 9 \\ \Rightarrow & b=4 & a=1, c=3 \\ & n(A \cup B)=5+7-4=8\end{aligned}$$