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 \( V=\{a, e, i, o, u\} \). (a) Find \( P \cup V \). (b) Find \( n(P \cap V) \). (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 \( P^{\prime} \).

To tackle this problem, let’s break it down piece by piece while keeping it dynamic and engaging! For part (a), we want to find the union of sets \( P \) and \( V \). This is like mixing two color palettes together to create a wider range of hues! So, when we combine the letters in both sets, we get: \( P \cup V = \{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}, \mathrm{i}, \mathrm{j}, u\} \). In part (b), we need to determine how many letters are common between \( P \) and \( V \). Let’s look for those consonant and vowel connections! The only matching vowels between these sets are \( \mathrm{a} \) and \( \mathrm{e} \), so: \( n(P \cap V) = 2 \). Moving on to part (c), we are asked for the maximum size of the subset \( L \) under the conditions that \( L \) is a subset of both \( P \) and \( V \). Given that \( V \) contains vowels and \( P \) has consonants and vowels, the only possible letters in \( L \) can be from the intersection. Therefore, the maximum value of \( n(L) \) would be limited to the two vowels we found earlier, giving us \( \mathrm{n}(L)_{max} = 2 \). Lastly, for part (d), we want to find the probability that a letter chosen at random from \( V \) is also in \( P^{\prime} \) (the complement of \( P \)). The letters in \( V \) are \( \{a, e, i, o, u\} \). The complement \( P^{\prime} \) contains all letters not in \( P \). Notably, in \( V \), \( a \) and \( e \) are in \( P \), so the letters \( i, o, \) and \( u \) are in \( P^{\prime} \). Therefore, the probability is \( \frac{3}{5} \), as you have 3 suitable letters out of 5 total in \( V \). And there you have it! How's that for squishing some math into a fun little package? 🎉