Question: (ii) blue on the spinner and an even number on the die.
Question 5 (Suggested maximum time: 5 minutes)
Let $U = \{x \in \mathbb{N}: 1 \leq x \leq 30\}$
$P = \{x \in U: x \text{ is a multiple of } 4\}$
$Q = \{x \in U: x \text{ is a multiple of } 6\}$
(a) Represent $U$, $P$ and $Q$ on the following Venn diagram.
Venn diagram shape: a rectangle labeled $U$ contains two overlapping circles, the left circle labeled $P$ and the right circle labeled $Q$, positioned center on the page.
(b) Use your Venn diagram to prove the following identity is true:
$\#(P \cup Q) = \#P + \#Q - \#(P \cap Q)$
(c) Use your Venn diagram to investigate if the following identity is true or false:
$\#(P \cup Q)' = \#(P' \cap Q')$
1. **State the problem:**
We have a universal set $U = \{x \in \mathbb{N}: 1 \leq x \leq 30\}$, and subsets $P$ and $Q$ where $P$ is multiples of 4 and $Q$ is multiples of 6 within $U$.
2. **Find elements of $P$ and $Q$:**
- Multiples of 4 in $U$: $P = \{4, 8, 12, 16, 20, 24, 28\}$
- Multiples of 6 in $U$: $Q = \{6, 12, 18, 24, 30\}$
3. **Find intersection $P \cap Q$:**
- Common multiples of 4 and 6 are multiples of their least common multiple, 12.
- So $P \cap Q = \{12, 24\}$
4. **Count elements:**
- $\#P = 7$
- $\#Q = 5$
- $\#(P \cap Q) = 2$
5. **Calculate $\#(P \cup Q)$:**
- By the formula for union:
$$\#(P \cup Q) = \#P + \#Q - \#(P \cap Q) = 7 + 5 - 2 = 10$$
6. **Verify by listing $P \cup Q$:**
- $P \cup Q = \{4, 6, 8, 12, 16, 18, 20, 24, 28, 30\}$
- Count: 10 elements, matches calculation.
7. **Prove identity (b):**
- The identity $\#(P \cup Q) = \#P + \#Q - \#(P \cap Q)$ holds true as shown.
8. **Investigate identity (c):**
- Complement sets relative to $U$:
- $P' = U \setminus P$
- $Q' = U \setminus Q$
- By De Morgan's law:
$$ (P \cup Q)' = P' \cap Q' $$
- Therefore, the cardinalities satisfy:
$$ \#(P \cup Q)' = \#(P' \cap Q') $$
- This identity is true by set theory.
**Final answers:**
- (b) Identity is true.
- (c) Identity is true.