Suppose $P$ and $Q$ are sets. Find the number of elements of the set $Q$ if you know that: $$|P\cap Q| = 63,\quad |P \cup Q| = 441,\quad |P| = 154.$$
Joseph solved the problem in the following steps:
(1) He drew a Venn diagram, marked the sets and their individual subsets in it:
(2) According to the assignment, Joseph set up a system of equations: \begin{aligned} b &= 63\cr a + b + c &= 441\cr a + b &= 154\cr c &= ? \end{aligned}
(3) He calculated the value of $c$: $$154 + c = 441 \Rightarrow c = 287$$
(4) Finally, he made a conclusion: $|Q| = 287$.
Is Joseph's solution correct? If not, determine where Joseph made a mistake.
Joseph's solution is correct.
The mistake is in step (2). According to the assignment, Joseph set up the system of equations incorrectly.
The mistake is in step (3). Joseph constructed the system of equations correctly. However, he incorrectly calculated its solution.
The mistake is in step (4). Joseph's conclusion about the number of elements of the set $Q$ is incorrect.
Joseph made the incorrect conclusion about the number of elements of the set $Q$. He determined the number of elements that are only in the set $Q$ and are not in the other set. The total number of elements of the set $Q$ is $|Q| = b + c = 63 + 287 = 350$.
