18. It is given that \( \xi=\{ \) letters of the alphabet \( \} \), \( P=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}, \mathrm{i}, \mathrm{j}\} \) and \( V=\{\mathrm{a}, \mathrm{e}, \mathrm{i}, \mathrm{o}, \mathrm{u}\} \). (a) Find \( P \cup V \). (b) Find \( \mathrm{n}\left(P \cap V^{\prime}\right) \). (c) Given that \( L \subset P \) and \( L \subset V \), state the maximum value of \( \mathrm{n}(L) \). (d) A letter is picked at random from \( V \). Find the probability that it is an element of \( P^{\prime} \).

The union of sets \( P \) and \( V \) combines all unique elements from both sets. So, \( P \cup V = \{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}, \mathrm{i}, \mathrm{j}, \mathrm{o}, \mathrm{u}\} \). To find \( n(P \cap V') \), we need to determine which elements in \( P \) are not in \( V \). The complement \( V' = \{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{f}, \mathrm{g}, \mathrm{h}, \mathrm{j}\} \). The intersection \( P \cap V' = \{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{f}, \mathrm{g}, \mathrm{h}, \mathrm{j}\} \) gives us \( n(P \cap V') = 7 \). Since \( L \) must be a subset of both \( P \) and \( V \), and the only overlapping elements in both are \( \mathrm{a}, \mathrm{e}, \mathrm{i} \) (which are vowels), the maximum value of \( n(L) \) is 3. When picking a letter from \( V = \{\mathrm{a}, \mathrm{e}, \mathrm{i}, \mathrm{o}, \mathrm{u}\} \), we see that \( P' \) contains letters not in \( P \), specifically \( \{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{f}, \mathrm{g}, \mathrm{h}, \mathrm{j}, \mathrm{o}, \mathrm{u}\} \). Hence, the probability of selecting a letter from \( P' \) is calculated as follows: there are 2 letters in \( V \) that are in \( P' \) (namely \( \mathrm{o} \) and \( \mathrm{u} \)), out of 5 total letters in \( V \), so the probability is \( \frac{2}{5} \).