Question 1.
For each of the following sets, decide whether it is finite, countable, or uncountable. Explain your answer briefly.
(1) $P(Q)$
Final Answer: The set $P(Q)$ is \boxed{\text{uncountable}}
Step 1 :The set $P(Q)$ refers to the power set of the set of rational numbers. The power set of a set is the set of all subsets of the set.
Step 2 :The cardinality of the power set of a set with cardinality $n$ is $2^n$.
Step 3 :The set of rational numbers is countably infinite, which means it has the same cardinality as the set of natural numbers.
Step 4 :Therefore, the power set of the set of rational numbers has a cardinality of $2^{\aleph_0}$, where $\aleph_0$ is the cardinality of the set of natural numbers.
Step 5 :This is the cardinality of the set of real numbers, which is uncountable.
Step 6 :Therefore, the set $P(Q)$ is uncountable.
Step 7 :Final Answer: The set $P(Q)$ is \boxed{\text{uncountable}}