Question

From Venn diagram show that $n\left( A\cup B \right)+n\left( A\cap B \right)=n\left( A \right)+n\left( B \right)$.

Answer 1

In this question we have been given with a Venn diagram and using the Venn diagram we have to show that $n\left( A\cup B \right)+n\left( A\cap B \right)=n\left( A \right)+n\left( B \right)$. We will first write all the sets which include the intersection and union set of the given sets $A$ and $B$. We will then consider the left-hand side of the expression and find its value and then consider the right-hand side and find its value and check whether the condition holds true.

Complete step by step answer:
We can see from the Venn diagram that:
\Rightarrow A=\left\\{ a,b,c,d \right\\}
\Rightarrow B=\left\\{ a,b,e,f,g \right\\}
Therefore, we get:
$\Rightarrow n\left( A \right)=4$
$\Rightarrow n\left( B \right)=5$
The union set which is represented as $A\cup B$ represents the set of all the distinct elements that are present in the sets $A$ or $B$. We have the intersection as:
\Rightarrow A\cup B=\left\\{ a,b,c,d,e,f,g \right\\}
Therefore, we have:
$\Rightarrow n\left( A\cup B \right)=7$
The intersection set which is represented as $A\cap B$ represents the set of all the distinct elements that are present in the both sets $A$ and $B$. We have the intersection as:
\Rightarrow A\cap B=\left\\{ a,b \right\\}
Therefore, we have:
$\Rightarrow n\left( A\cap B \right)=2$
Now we have to prove that $n\left( A\cup B \right)+n\left( A\cap B \right)=n\left( A \right)+n\left( B \right)$.
Consider the left-hand side of the expression, we have:
$\Rightarrow n\left( A\cup B \right)+n\left( A\cap B \right)$
On substituting the values, we get:
$\Rightarrow 7+2$
On simplifying, we get:
$\Rightarrow 9$, which is the value of the left-hand side.
Consider the right-hand side of the expression, we have:
$\Rightarrow n\left( A \right)+n\left( B \right)$
On substituting the values, we get:
$\Rightarrow 4+5$
On simplifying, we get:
$\Rightarrow 9$, which is the value of the right-hand side.
Since the values of both the sides are the same the expression is correct, hence proved.

Note: It is to be remembered that the $n\left( P \right)$ represents the number of elements present in the set. A Venn diagram is a good method to simplify set problems such that they can be visualized. The various properties of Venn diagrams should be remembered to solve these types of questions.