Q73 / 155 · WBJEE 2023 · Mathematics
Let \(A, B, C\) are subsets of set \(X\). Then consider the validity of the following set theoretic statement :
- A\(A \cup(B \backslash C)=(A \cup B) \backslash(A \cup C)\)
- Correct answerB\((A \backslash B) \backslash C=A \backslash(B \cup C)\)
- C\((A \cup B) \backslash A=A \backslash B\)
- D\(A \backslash C=B \backslash C\)
Solution
Hint:
\(\begin{aligned} :(A \backslash B) \backslash C & =\left(A \cap B^{\prime}\right) \cap C^{\prime} \\ & =A \cap\left(B^{\prime} \cap C^{\prime}\right) \\ & =A \cap(B \cup C)^{\prime} \\ & =A \backslash(B \cup C) \end{aligned}\)
\(\begin{aligned} :(A \backslash B) \backslash C & =\left(A \cap B^{\prime}\right) \cap C^{\prime} \\ & =A \cap\left(B^{\prime} \cap C^{\prime}\right) \\ & =A \cap(B \cup C)^{\prime} \\ & =A \backslash(B \cup C) \end{aligned}\)
Hence, the correct answer is option: 'B'.