For any three sets A, B, and C prove that: $A\cap \left( B\cup C \right)=\left( A\cap B \right)\cup \left( A\cap C \right)$

Hint:First we will draw the Venn diagram and then we will divide all the regions into different numbers and then we will use these numbers to perform all the operations like union and intersection and then we will be prove $A\cap \left( B\cup C \right)=\left( A\cap B \right)\cup \left( A\cap C \right)$.

Complete step-by-step answer:
Let’s look at the Venn diagram for LHS,
First we will look at $\left( B\cup C \right)$,

Now $A\cap \left( B\cup C \right)$ will be,

Now for the RHS, we will draw $\left( A\cap B \right)$

And this is $\left( A\cap C \right)$,

Now $\left( A\cap B \right)\cup \left( A\cap C \right)$ will be,

Let’s first draw the required Venn diagram,

As we can see that we have divided the regions into 7 parts.
Keep in mind that the numbers that we are going to use are the regions.
Here intersection of sets means the common region or common numbers to be more clear.
And union means adding all the different regions in the two sets.
Now we will use this to specify all the required relations,
A = 1 + 4 + 5 + 7
B = 2 + 5 + 6 + 7
C = 3 + 4 + 6 + 7
Let’s first solve RHS,
Now we will find $A\cap B$, we have A = 1 + 4 + 5 + 7 and B = 2 + 5 + 6 + 7
The common regions between A and B are 5 and 7.
Hence,
$A\cap B$ = 5 + 7
Now we will find $A\cap C$, we have A = 1 + 4 + 5 + 7 and C = 3 + 4 + 6 + 7
The common regions between A and C are 4 and 7.
Hence,
$A\cap C$ = 4 + 7
Now we have $A\cap B$ = 5 + 7 and $A\cap C$ = 4 + 7, now taking all the regions in these two sets we get,
$\left( A\cap B \right)\cup \left( A\cap C \right)$ = 4 + 5 + 7
Now let’s solve LHS,
Now we will find $B\cup C$,
B = 2 + 5 + 6 + 7 and C = 3 + 4 + 6 + 7, now taking all the regions in B and C we get,
$B\cup C$ = 2 + 5 + 7 + 6 + 4 + 3
Now we have A = 1 + 4 + 5 + 7 and $B\cup C$ = 2 + 5 + 7 + 6 + 4 + 3, the common region between these two is 4, 5, and 7.
Therefore,
$A\cap \left( B\cup C \right)$ = 4 + 5 + 7
Hence, LHS = RHS.
Hence we have proved that $A\cap \left( B\cup C \right)=\left( A\cap B \right)\cup \left( A\cap C \right)$.

Note: Keep in mind that the numbers that we have used should not be added as integers because they are the regions that we have used in the Venn diagram. Here one can also solve this question by explaining the meaning of all the operations that we have used and show that in the end they will be equal, but the method that we have used is more clear and helps one understand the solution better.