For any sets A, B, C prove that:
$\left( A\cap B \right)\cap C=A\cap \left( B\cap C \right)$ [Associative law of intersection of sets]

Hint: Here, to prove $\left( A\cap B \right)\cap C=A\cap \left( B\cap C \right)$, we have to prove $\left( A\cap B \right)\cap C\subset A\cap \left( B\cap C \right)$ and $A\cap \left( B\cap C \right)\subset \left( A\cap B \right)\cap C$. For the first part take $x\in \left( A\cap B \right)\cap C$ and then we have to show that $x\in A\cap (B\cap C)$. For the second part we have to take $x\in A\cap \left( B\cap C \right)$ and to show that $x\in \left( A\cap B \right)\cap C$.

Complete step-by-step answer:
Here, we have to prove that $\left( A\cap B \right)\cap C=A\cap \left( B\cap C \right)$.
Here, we have to show that $\left( A\cap B \right)\cap C\subset A\cap \left( B\cap C \right)$ and $A\cap \left( B\cap C \right)\subset \left( A\cap B \right)\cap C$ which will give $\left( A\cap B \right)\cap C=A\cap \left( B\cap C \right)$

First consider LHS, $\left( A\cap B \right)\cap C$

Let $x\in \left( A\cap B \right)\cap C$ then we can say that $x\in A\cap B$ and $x\in C$

$\Rightarrow x\in $ (A and B) and $x\in C$

By $x\in $ (A and B) we can say that $x\in A$ and $x\in B$.

Therefore, we can write:

$x\in A$, $x\in B$and $x\in C$

$\Rightarrow x\in A$, $x\in $ (B and C)

$\Rightarrow x\in A$, and (B and C)

$\Rightarrow x\in A\cap (B\cap C)$

Hence, we can say that,

$\left( A\cap B \right)\cap C\subset A\cap \left( B\cap C \right)$ ….. (1)

Now, consider RHS $A\cap \left( B\cap C \right)$.

Let $x\in A\cap \left( B\cap C \right)$ then we can say that $x\in A$ and $x\in B\cap C$

$\Rightarrow x\in A$ and $x\in $ (B and C)

By $x\in $ (B and C) we can say that $x\in B$ and $x\in C$.

Therefore, we can write:

$x\in A$, and $x\in B$, $x\in C$

$\Rightarrow x\in $ (A and B) and $x\in C$

$\Rightarrow x\in $ (A and B) and C

$\Rightarrow x\in (A\cap B)\cap C$

Hence, we can say that,

$A\cap \left( B\cap C \right)\subset \left( A\cap B \right)\cap C$ ….. (2)

Now, from equation (1) and equation (2) we can say that,

$\left( A\cap B \right)\cap C=A\cap \left( B\cap C \right)$.


Note: Here, $\left( A\cap B \right)\cap C=A\cap \left( B\cap C \right)$ is called the associative law of intersection of sets. We can also prove this with the help of Venn diagram, by taking the three sets A, B and C. Two diagrams are required, one for $\left( A\cap B \right)\cap C$ and the other for $A\cap \left( B\cap C \right)$. Thus, we have to shade the required regions which will be the same.