I've tried to solve the question in two different ways :
- Using Characteristic function(CF).
$$CF(A) = \{ 1, x \in A \text{ ; } 0, \text{elsewhere}\}$$ $$CF(B) = \{ 1, x \in B \text{ ; } 0, \text{elsewhere} \}$$ $$CF(C) = \{ 1, x \in C \text{ ; } 0, \text{elsewhere} \}$$
Since, $$A \cap C = B \cap C $$ $$CF(A) * CF(B) = CF(B) * CF (C)$$ $$\text{Therefore, } CF(A) = CF(B)$$ $$\implies A=B$$
Consider the following counter example :
Let $$A = \{1,2,3\} , B = \{2,3,4,5\} , C = \{2,3\}.$$ $$A \cap C = \{2,3\} \text{ and } B \cap C = \{2,3\} \text{ but } A \neq B.$$ Hence disproved by counter example.
Which one is correct ? Many thanks in advance.