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 = B
(B). A = C
(C). B = C
(D). $A \cap B = \phi $

Answer 1

Hint: Before attempting this question one must have prior knowledge about the concept of sets and also remember that if $A \cup B = A \cup C$ which means all elements of B are in set A and set C, using this information will help you to approach the solution of the question.


Complete step-by-step answer:
According to the given information it is given that $A \cap B = A \cap C$ and $A \cup B = A \cup C$
so, the set of B is given as; $B = \left( {A \cup B} \right) \cap B$
since, $A \cup B = A \cup C$
Therefore, $B = \left( {A \cup C} \right) \cap B$
As we know that by the distributive property i.e. $\left( {A \cup C} \right) \cap B = \left( {A \cap B} \right) \cup \left( {B \cap C} \right)$
Therefore, $B = \left( {A \cap B} \right) \cup \left( {B \cap C} \right)$
Since, $A \cap B = A \cap C$
So, $B = \left( {A \cap C} \right) \cup \left( {B \cap C} \right)$
Also, we know that by the distributive property i.e.$\left( {A \cup B} \right) \cap C = \left( {A \cap C} \right) \cup \left( {B \cap C} \right)$
Therefore, $B = \left( {A \cup B} \right) \cap C$
Since, $A \cup B = A \cup C$
Therefore, $B = \left( {A \cup C} \right) \cap C$
Since, $\left( {A \cup C} \right) \cap C = C$
Therefore, $B = C$
Hence, option C is the correct option.

Note: In the above solution we used the term “set” which can be explained as an organized manner of collections of objects or elements which are represented as set-builder form or a roster form, generally the representation of sets is given as {}. In the sets numbers of elements and size are identified by the order of sets which is named as cardinality.