Question

If X and Y are two sets such that $X \cup Y$ has $60$ elements, $X$ has $33$ elements and $Y$ has $37$ elements, how many elements does $X - Y$ and $Y - X$ have?

Answer 1

Hint: We will use the formula: $n\left( {A - B} \right) = n\left( {A \cup B} \right) - n\left( B \right)$ to calculate the number of elements in $X - Y$ and $X - Y$. Here, we will suppose $X \equiv A{\text{ and }}Y \equiv B$ and the number of elements of ${\text{X and }}Y$ are already given in the question. After putting their values, we will get the number of elements in $X - Y$ and $Y - X$.

Complete step-by-step answer:
We are given that $X$ and $Y$ are two sets such that $X \cup Y$ has $60$ elements, $X$ has $33$ elements and $Y$ has $37$ elements.
We are required to calculate the number of elements in $X - Y$ and $Y - X$.
We have the formula of sets given by: $n\left( {A - B} \right) = n\left( {A \cup B} \right) - n\left( B \right)$
Here, let $A = X$ and $B = Y$.
Now, the number of elements in the set $X$ are given as: $n\left( X \right) = 33$.
The number of elements in the set $Y$can be represented as: $n\left( Y \right) = 37$.
The number of elements in the union of both the sets $X{\text{ and }}Y$ can be given by: $n\left( {X \cup Y} \right) = 60$.
So, for calculating the number of elements in the set $X - Y$, we can write the formula as:
$ \Rightarrow n\left( {X - Y} \right) = n\left( {X \cup Y} \right) - n\left( Y \right)$
On substituting the values, we get
\[
   \Rightarrow n\left( {X - Y} \right) = 60 - 37 \\
   \Rightarrow n\left( {X - Y} \right) = 23 \\
 \]
Therefore, there are a total of $23$ elements in the set \[X - Y\].
Now, for the calculation of the number of elements in $Y - X$, we have the formula as: $n\left( {Y - X} \right) = n\left( {X \cup Y} \right) - n\left( X \right)$
On substituting the values, we get
$
   \Rightarrow n\left( {Y - X} \right) = 60 - 33 \\
   \Rightarrow n\left( {Y - X} \right) = 27 \\
 $
Hence, there are a total $27$ elements in the set $Y - X$.

Note: We can also solve this question using the other formula: \[n\left( {A - B} \right) = n\left( A \right) - n\left( {A \cap B} \right)\] and $n\left( {A \cap B} \right)$ can be calculated using the formula: $n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)$. This is a time consuming method, comparatively.