Question

If X and Y are two sets such that X has 40 elements, $X\cup Y$ has 60 elements and $X\cap Y$ has 10 elements, how many elements does Y have?

Answer 1

Hint: If X and Y are two sets then, there is a relation between $n\left( X \right),n\left( Y \right),n\left( X\cup Y \right),n\left( X\cap Y \right)$ where n (A) = number of elements in set A. The formula is given as;
\[n\left( X\cup Y \right)=n\left( X \right)+n\left( Y \right)-n\left( X\cap Y \right)\]
We will substitute the given values to get n (Y) that is the number of elements in Y.

Complete step-by-step answer:
Let A be a set then n (A) represents the number of elements in set A.
We are given X has 40 elements, $\Rightarrow n\left( X \right)=40$
Given $X\cup Y$ has 60 elements $\Rightarrow n\left( X\cup Y \right)=60$
And $X\cap Y$ has 10 elements $\Rightarrow n\left( X\cap Y \right)=40$
Before solving further let us first understand what are $X\cup Y\text{ and }X\cap Y$
\[\begin{align}
  & X\cup Y=\text{ union of X and Y} \\
 & X\cap Y=\text{ intersection of X and Y} \\
\end{align}\]
Let us define union and intersection of two sets.
The union of two sets is a new set that contains all elements that are in at least one of the two sets. It is represented as $A\cup B$ where A and B are sets.
The intersection of two sets is a new set that contains all elements that are common in both sets. It is represented as $A\cap B$ where A and B are sets.
So, we have \[\Rightarrow n\left( X\cup Y \right)=60\text{ and }n\left( X\cap Y \right)=10\]
We have to calculate number of elements in Y $\Rightarrow n\left( Y \right)=?$
We will use a formula to calculate this, given as below:
\[n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)\]
Using X and Y in place of A and B we get:
\[n\left( X\cup Y \right)=n\left( X \right)+n\left( Y \right)-n\left( X\cap Y \right)\]
Substituting values in above we get:
\[\begin{align}
  & 60=40+n\left( Y \right)-10 \\
 & 60-40+10=n\left( Y \right) \\
 & n\left( Y \right)=70-40 \\
 & n\left( Y \right)=30 \\
\end{align}\]
Therefore, the number of elements in Y is 30.

Note: Another way to solve this question is by using Venn diagram;
We have \[\begin{align}
  & n\left( X \right)=40 \\
 & n\left( Y \right)=? \\
 & n\left( X\cup Y \right)=60 \\
\end{align}\]
Let us use all these values in Venn diagram given below:

Clearly by Venn diagram we observe that, number of elements in \[X\text{ only}=n\left( X \right)-n\left( X\cap Y \right)\]
Number of elements in \[\text{X only}=40-10=30\]
\[\Rightarrow \text{X only}=30\]
By Venn diagram we have:
\[\begin{align}
  & n\left( Y \right)=n\left( X\cup Y \right)-n\left( X\text{ only} \right) \\
 & n\left( Y \right)=60-30=30 \\
 & \Rightarrow n\left( Y \right)=30 \\
\end{align}\]
Therefore, n (Y) = 30 which is the required result.