Question

If $X$ and $Y$ are two sets such that $n\left( X \right) = 17$ ,$n\left( Y \right) = 23$ and $n\left( {X \cup Y} \right) = 38$. Find $n\left( {X \cap Y} \right)$.

Answer 1

Hint: In the question, we have given a number of elements in two sets and the number of elements in the union of the two sets. We need to find the intersection of the two sets. We will use operations applied to set theory. According to these theories, the number of elements in the union of sets can be found by subtracting the number of elements in the intersection of the sets to the summation of the number of elements in the individual sets.

Complete step-by-step answer:
The relation between the number of elements in set $X$ , $Y$, the intersection of sets and union of sets can be expressed as:
$\Rightarrow n\left( {X \cup Y} \right) = n\left( X \right) + n\left( Y \right) - n\left( {X \cap Y} \right)$
Given: $X$ and $Y$ are two sets
$\Rightarrow$ The number of elements in set $X$ are $n\left( X \right) = 17$
The number of elements in set $Y$ are $n\left( Y \right) = 23$
The union of the sets $X$ and $Y$ are $n\left( {X \cup Y} \right) = 38$

We need to find the union of the sets $X$ and $Y$ which can be found out by the formula given by the set theory.
We know that the relation between the number of elements in set $X$ , $Y$, the intersection of sets and union of sets can be expressed as:
$\Rightarrow n\left( {X \cup Y} \right) = n\left( X \right) + n\left( Y \right) - n\left( {X \cap Y} \right)$
We will substitute 17 for $n\left( X \right)$ , 23 for $n\left( Y \right)$ and 38 for $n\left( {X \cup Y} \right)$ in the above expression.
 \[
\Rightarrow 38 = 17 + 23 - n\left( {X \cap Y} \right)\\
\Rightarrow n\left( {X \cap Y} \right) = 40 - 38\\
\Rightarrow n\left( {X \cap Y} \right) = 2
\]
Hence, the intersection of the sets $X$ and $Y$ is \[n\left( {X \cap Y} \right) = 2\].

Note: Set theory deals with a well-defined collection of data of both mathematical nature or non-mathematical nature. In set theory, we define the union of the sets as the collection of all the elements present in all the sets in consideration. Also, the intersection of the sets can be defined as the number of elements that are common in the different sets. We generally use the Venn diagram to solve questions related to sets.