Question
If $$A, B$$ and $$C$$ are three sets such that $$A \cap B = A \cap C$$ and $$A \cup B = A \cup C,$$ then
A.
$$A = C$$
B.
$$B = C$$
C.
$$A \cap B = \phi $$
D.
$$A = B$$
Answer :
$$B = C$$
Solution :
Let $$x \in A\,\,{\text{and }}x \in B$$
$$\eqalign{
& \Leftrightarrow \,x \in A \cup B \cr
& \Leftrightarrow \,\,x \in A \cup C\,\,\,\left( {\because A \cup B = A \cup C} \right) \cr
& \Leftrightarrow x \in C \cr
& \therefore \,\,B = C. \cr} $$
Let $$x \in A\,\,{\text{and }}x \in B$$
$$\eqalign{
& \Leftrightarrow \,x \in A \cap B \cr
& \Leftrightarrow \,\,x \in A \cap C\,\,\,\left( {\because A \cap B = A \cap C} \right) \cr
& \Leftrightarrow x \in C \cr
& \therefore \,\,B = C. \cr} $$