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 \( P^{\prime} \). (a) Given that \( \xi=\{1,2,3,4, \ldots, \mathrm{t}\} \) where t is a positive integer, \( P \) and \( Q \) are two subsets of \( \xi \) where \( P=\{x: x \) is a prime number \( \} \) and \( Q=\{x: \) the unit digit of \( x \) is 7\( \} \). If \( \mathrm{n}(P \cap Q)=5 \), write down the minimum value of t . (b) Two sets \( A \) and \( B \) are such that \( \mathrm{n}(A)=32 \) and \( \mathrm{n}(B)=19 \). Given n( \( \xi)=50, B \not \subset A \) and ( \( (A \cup B)^{\prime}=8 \), find the value of \( \mathrm{n}(A \cap B) \).

**(i) For the sets defined on the alphabet:** 1. \(P=\{a,b,c,d,e,f,g,h,i,j\}\) \(V=\{a,e,i,o,u\}\) 2. **(a) Find \(P\cup V\):** The union \(P\cup V\) is the set of letters that are in either \(P\) or \(V\): \[ P\cup V=\{a,b,c,d,e,f,g,h,i,j,o,u\} \] 3. **(b) Find \( \mathrm{n}(P\cap V') \):** Here, \(V'\) is the complement of \(V\) (with respect to the universal set \(\xi\), the set of all letters of the alphabet). Thus, \(P\cap V'\) consists of the letters in \(P\) that are not vowels. In \(P\), the vowels are \(a\), \(e\), and \(i\). Removing these from \(P\) gives: \[ P\cap V'=\{b,c,d,f,g,h,j\} \] So, \[ \mathrm{n}(P\cap V')=7. \] 4. **(c) Given \(L\subset P\) and \(L\subset V\), state the maximum value of \(\mathrm{n}(L)\):** For \(L\) to be a subset of both \(P\) and \(V\), it must be contained in their intersection: \[ P\cap V=\{a,e,i\} \] Hence, the maximum number of elements \(L\) can have is: \[ \mathrm{n}(L)=3. \] 5. **(d) A letter is picked at random from \(V\). Find the probability that it is an element of \(P'\):** \(P'\) is the complement of \(P\) (with respect to \(\xi\)). \(V=\{a,e,i,o,u\}\) and note that \[ P=\{a,b,c,d,e,f,g,h,i,j\}. \] The letters in \(V\) that are in \(P\) are \(a\), \(e\), and \(i\). Thus, the letters in \(V\) that are not in \(P\) are: \[ \{o,u\}. \] The probability is therefore: \[ \text{Probability}=\frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}=\frac{2}{5}. \] --- **(ii) For the sets defined on the set \(\xi=\{1,2,3,\ldots,t\}\):** 1. **(a)** \(P=\{x: x \text{ is a prime number}\}\) and \(Q=\{x: \text{the unit digit of } x \text{ is } 7\}\). We are given that \[ \mathrm{n}(P\cap Q)=5. \] List the smallest prime numbers ending with 7: \[ 7,\; 17,\; 37,\; 47,\; 67,\; 97,\ldots \] The five smallest primes are \(7\), \(17\), \(37\), \(47\), and \(67\). To have precisely these five in \(P\cap Q\), the set \(\xi\) must include \(67\) but not \(97\). Thus, the minimum value of \(t\) is \[ t=67. \] 2. **(b)** We are given two sets \(A\) and \(B\) with \[ \mathrm{n}(A)=32,\quad \mathrm{n}(B)=19,\quad \mathrm{n}(\xi)=50, \] and \[ \mathrm{n}((A\cup B)')=8. \] Since \((A\cup B)'\) has 8 elements, then \[ \mathrm{n}(A\cup B)=\mathrm{n}(\xi)-8=50-8=42. \] Using the formula for the union of two sets, we have: \[ \mathrm{n}(A\cup B)=\mathrm{n}(A)+\mathrm{n}(B)-\mathrm{n}(A\cap B). \] Substitute the known values: \[ 42=32+19-\mathrm{n}(A\cap B). \] This simplifies to: \[ 42=51-\mathrm{n}(A\cap B) \quad\Longrightarrow\quad \mathrm{n}(A\cap B)=51-42=9. \] The condition \(B\not\subset A\) is satisfied since \(\mathrm{n}(A\cap B)=9<\mathrm{n}(B)=19\). --- **Final Answers:** 1. \(P\cup V=\{a,b,c,d,e,f,g,h,i,j,o,u\}\) 2. \(\mathrm{n}(P\cap V')=7\) 3. Maximum \(\mathrm{n}(L)=3\) 4. The probability is \(\frac{2}{5}\) 5. Minimum \(t=67\) 6. \(\mathrm{n}(A\cap B)=9\)