Question

If V=\left\\{ a,e,i,o,u \right\\} and B=\left\\{ a,i,k,u \right\\}. Then find $V-B$ and $B-V$. Are they equal?

Answer 1

In this question we have been given with two sets $V$ and $B$ which as elements in them. We have been told to find the set operation $V-B$ and $B-V$. The set operation $V-B$ represents all the elements of $V$ which are not present in $B$ and the set operation $B-V$ represents all the elements of $B$ which are not present $V$. We will first find both the set operations and then check whether both the sets are equal or not and get the required solution.

Complete step by step solution:
We have the set $V$ given to us as:
\Rightarrow V=\left\\{ a,e,i,o,u \right\\}
We have the set $B$ given to us as:
\Rightarrow B=\left\\{ a,i,k,u \right\\}
Now we have $V-B$ as all the elements present in $V$ which are not present in $B$. We can see that the elements $e$ and $o$ are only present in $V$ and not in $B$, therefore, we can write:
\Rightarrow V-B=\left\\{ e,o \right\\}
Now we have $B-V$ as all the elements present in $B$ which are not present in $V$. We can see that the element $k$ only present in $B$ and not in $V$, therefore, we can write:
\Rightarrow B-V=\left\\{ k \right\\}
Now on comparing the sets $V$ and $B$, we can see that the sets are not the same therefore, we can conclude that $V-B$ and $B-V$ are not equal.

Note: It is to be remembered that two sets are equal when each element in one set is present in another set. This should not be considered as the equivalency of sets. Two sets are said to be equivalent when the number of elements present in both the set are the same. Mathematically $A=B$ when $A\subset B$ and $B\subset A\Leftrightarrow A=B$. And two sets are equivalent when $n\left( A \right)=n\left( B \right)$.