Let $A=\left\{ a,e,i,o,u \right\}\text{ and }B=\left\{ a,b,c,d \right\}$. Is A a subset of B? No. (Why?) Is B a subset of A? No. (Why?)

Hint: For solving such problems we must compare all the elements given in the set and then determine whether a set can be called as a subset or not.

Complete step-by-step answer:
A set is a well-defined collection of objects. Here the term ‘well-defined’ means that there is some definite rule on the basis of which one can decide whether an object is in the collection or not. Sets are generally denoted by capital letters as we have seen in our question also. If an object ‘a’ is in set A, then we can say that ‘a’ is an element of set A or ‘a’ belongs to set A$\left( a\in A \right)$. If an object ‘a’ is not in set A then we can say ‘a’ does not belong to set A$\left( a\notin A \right)$.
There are many types of sets but in this problem, we are given a finite set. A finite set is the one having some finite, countable number of elements.
Now proceeding to the definition of subset.
Let two sets P and Q have some finite number of elements. If every element of set P is an element of set Q, then P is said to be the subset of Q and we express it as $P\subseteq Q$.
Now the two sets given are:
$A=\left\{ a,e,i,o,u \right\}$ and $B=\left\{ a,b,c,d \right\}$.
For the first part we compare each element of set A with set B and found that elements (e, i, o, u) are missing in set B. Hence, we concluded that A is not a subset of B, $\therefore A\not\subset B$.
For the second part we compare each element of set B with set A and found that elements (b, c, d) are missing in set A. Hence, $B\not\subset A$.
So, none of the sets is a subset amongst the given set.

Note: The key tip for solving this problem is checking each element step by step according to the definition of subset of any set.