Which expression is equal to \(A\setminus(B\cap C)\)?
Answer and explanation
Correct answer: \((A\setminus B)\cup(A\setminus C)\)
Use the difference identity \(A\setminus X=A\cap X'\). Thus \(A\setminus(B\cap C)=A\cap(B\cap C)'\). By De Morgan's law, \((B\cap C)'=B'\cup C'\), so the expression becomes \(A\cap(B'\cup C')\). Distributing intersection over union gives \((A\cap B')\cup(A\cap C')\), which is exactly \((A\setminus B)\cup(A\setminus C)\). Therefore option A is correct.
Frequently asked questions
What is the correct answer to this question?
\((A\setminus B)\cup(A\setminus C)\)
Why is this the correct answer?
Use the difference identity \(A\setminus X=A\cap X'\). Thus \(A\setminus(B\cap C)=A\cap(B\cap C)'\). By De Morgan's law, \((B\cap C)'=B'\cup C'\), so the expression becomes \(A\cap(B'\cup C')\). Distributing intersection over union gives \((A\cap B')\cup(A\cap C')\), which is exactly \((A\setminus B)\cup(A\setminus C)\). Therefore option A is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
