Show That A ∩ B = A ∩ C Need Not Imply B = C

Solution

To show that the statement is not always true, we give a counterexample.

Let

\[ A=\{1\}, \quad B=\{1,2\}, \quad C=\{1,3\} \]

Now find \( A\cap B \):

\[ A\cap B=\{1\} \]

Next find \( A\cap C \):

\[ A\cap C=\{1\} \]

Therefore,

\[ A\cap B=A\cap C \]

But,

\[ B=\{1,2\} \neq \{1,3\}=C \]

Hence,

\[ A\cap B=A\cap C \]

does not imply

\[ B=C \]

Hence proved.