If $A, B$ and $C$ are three sets such that $A \cap B=A \cap C$ and $A \cup B=A \cup C$, then :
JEE · Math · previous-year question
- A.$A=C$
- B.$B=C$correct
- C.$A \cap B=\phi$
- D.$A=B$
Answer
B. $B=C$
Explanation
From the given conditions, we have : A ∩ B = A ∩ C : The intersection of set A with set B is the same as the intersection of set A with set C. This indicates that all elements common to A and B are also common to A and C, and vice versa. A ∪ B = A ∪ C : The union of set A with set B is the same as the union of set A with set C. This indicates that all elements in A, B, and C are the same. From these two conditions, we can infer that set B is equal to set C because every element of B is also an element of C and vice versa. Hence, Option B : B = C is the correct answer.
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