1. **State the problem:** We have a universal set $U = \{x \in \mathbb{N} : 1 \leq x \leq 30\}$, and two subsets: - $P = \{x \in U : x \text{ is a multiple of } 4\}$ - $Q = \{x \in U : x \text{ is a multiple of } 6\}$ We want to: (a) Represent $U$, $P$, and $Q$ on a Venn diagram. (b) Prove the identity $\#(P \cup Q) = \#P + \#Q - \#(P \cap Q)$ using the Venn diagram. (c) Investigate if $\#(P \cup Q)' = \#(P' \cap Q')$ is true or false. 2. **Find the elements of each set:** - $P = \{4, 8, 12, 16, 20, 24, 28\}$ since these are multiples of 4 up to 30. - $Q = \{6, 12, 18, 24, 30\}$ since these are multiples of 6 up to 30. 3. **Find the intersection $P \cap Q$:** - $P \cap Q = \{12, 24\}$ (multiples of both 4 and 6). 4. **Count the elements:** - $\#P = 7$ - $\#Q = 5$ - $\#(P \cap Q) = 2$ 5. **Calculate $\#(P \cup Q)$:** - By listing elements in $P \cup Q$: $\{4, 6, 8, 12, 16, 18, 20, 24, 28, 30\}$ - So, $\#(P \cup Q) = 10$ 6. **Prove the identity $\#(P \cup Q) = \#P + \#Q - \#(P \cap Q)$:** - Substitute values: $$\#(P \cup Q) = 7 + 5 - 2 = 10$$ - This matches the counted value, so the identity is true. 7. **Investigate if $\#(P \cup Q)' = \#(P' \cap Q')$ is true:** - $P \cup Q$ has 10 elements, so its complement $(P \cup Q)'$ in $U$ has $30 - 10 = 20$ elements. - $P'$ is the complement of $P$ in $U$, so $\#P' = 30 - 7 = 23$. - $Q'$ is the complement of $Q$ in $U$, so $\#Q' = 30 - 5 = 25$. - $P' \cap Q'$ is the set of elements not in $P$ and not in $Q$, which is exactly the complement of $P \cup Q$. - Therefore, $\#(P' \cap Q') = \#(P \cup Q)' = 20$. **Conclusion:** The identity $\#(P \cup Q)' = \#(P' \cap Q')$ is true.