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 $$ P^{\prime} $$.

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**(a)**  
We have  
$$
P=\{a,b,c,d,e,f,g,h,i,j\} \quad \text{and} \quad V=\{a,e,i,o,u\}.
$$  
Taking the union, we include all elements in either $$ P $$ or $$ V $$:  
$$
P\cup V=\{a,b,c,d,e,f,g,h,i,j,o,u\}.
$$

**(b)**  
The complement $$ V' $$ (with respect to the alphabet) consists of all letters that are not vowels. Thus, $$ P\cap V' $$ contains those letters in $$ P $$ that are not vowels.  
Since  
$$
P=\{a,b,c,d,e,f,g,h,i,j\} \quad \text{and} \quad V=\{a,e,i,o,u\},
$$  
we have  
$$
P\cap V'=\{b,c,d,f,g,h,j\}.
$$  
The number of elements is  
$$
n(P\cap V')=7.
$$

**(c)**  
Given that $$ L\subset P $$ and $$ L\subset V $$, we have  
$$
L\subset P\cap V.
$$  
Since  
$$
P\cap V=\{a,e,i\},
$$  
the maximum number of elements in $$ L $$ is  
$$
n(L)=3.
$$

**(d)**  
A letter is picked at random from $$ V=\{a,e,i,o,u\} $$. We need the probability that it is an element of $$ P' $$ (the complement of $$ P $$ with respect to the alphabet).  
Checking each element in $$ V $$:  
- $$ a,e,i \in P $$  
- $$ o,u \notin P $$

Thus, there are 2 favorable outcomes, and the total number of outcomes is 5. Hence, the required probability is  
$$
\frac{2}{5}.
$$
**(a)** $$ P \cup V = \{a, b, c, d, e, f, g, h, i, j, o, u\} $$ **(b)** $$ n(P \cap V') = 7 $$ **(c)** The maximum value of $$ n(L) $$ is 3. **(d)** The probability is $$ \frac{2}{5} $$.

It is given that
letters of the alphabet ,

and .
(a) Find .
(b) Find .
© Given that and , state the maximum value of .
(d) A letter is picked at random from . Find the probability that it is an element of .

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It is given that letters of the alphabet , and . (a) Find . (b) Find . © Given that and , state the maximum value of . (d) A letter is picked at random from . Find the probability that it is an element of .