Question

Let A, B and C be the sets such that $ A \cup B = A \cup C $ and $ A \cap B = A \cap C $ . Show that \[B\; = C.\]

Answer 1

Hint: To solve this problem, we will first consider the given equation to us, i.e., $ A \cup B = A \cup C $ , by rearranging the equation, we will form two equations, in terms on B and in terms of C and then on taking both the equation, equals to each other, we will be able to show that \[B\; = C.\]

Complete step-by-step answer:
We have been given that A, B and C are the sets such that $ A \cup B = A \cup C $ and $ A \cap B = A \cap C $ . We need to show that \[B\; = C.\]
So, we have been given in the question that, $ A \cup B = A \cup C $
On taking intersection of C on both sides of the above equation, we get
\[\left( {A \cup B} \right) \cap C{\text{ }} = {\text{ }}\left( {A \cup C} \right) \cap C\]
Now, we know that, \[\left( {A \cup C} \right) \cap C{\text{ }} = {\text{ }}C\], so on putting this value in the above expression, we get
\[(A \cap C)\; \cup (B \cap C){\text{ }} = {\text{ }}C\;\]
Since, another condition is given in the question that, $ A \cap B = A \cap C $
Therefore, on putting the value in the above equation, we get
\[(A \cap B)\; \cup (B \cap C){\text{ }} = {\text{ }}C\;.......eq.(1)\]
Now, again we will take the given value, i.e., $ A \cup B = A \cup C $
On taking intersection of B on both sides, we get
\[\Rightarrow \left( {A \cup B} \right) \cap B{\text{ }} = {\text{ }}\left( {A \cup C} \right) \cap B\]
Now, we know that, \[\left( {A \cup B} \right) \cap B{\text{ }} = {\text{ B}}\] , so on putting this value in the above expression, we get
\[\Rightarrow B{\text{ }} = {\text{ }}\left( {A \cap B} \right)\; \cup \left( {C \cap B} \right)\]
We also know that, \[\left( {C \cup B} \right) = {\text{ (B}} \cup C),\] so on putting this value in the above expression, we get
\[\Rightarrow \;{\text{B}} = \left( {A \cap B} \right)\; \cup \left( {B \cap C} \right)\;\;{\text{ }}.......eq.(2)\]
So, from \[eq.\left( 1 \right)\] and \[\left( 2 \right),\] we get, \[B\; = C.\]
Thus, if $ A \cup B = A \cup C $ and $ A \cap B = A \cap C $ , then \[B\; = C.\]

Note: Students should note that in these types of questions, the mistake we always tend to make is by applying intersection sign in place of union sign and vice versa. So, always recheck after solving the question, as with one mistake a full solution can get wrong.