In a Venn diagram, what is (A ∪ B) ∩ (A ∪ Bᶜ) equal to?
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Answer and explanation
Correct answer: A
Apply the distributive identity \((X\cup Y)\cap(X\cup Z)=X\cup(Y\cap Z)\). Taking \(X=A\), \(Y=B\), and \(Z=B^c\), the expression becomes \(A\cup(B\cap B^c)\). A set and its complement are disjoint, so \(B\cap B^c=\varnothing\). Therefore the result is \(A\cup\varnothing=A\), making option A correct.
Frequently asked questions
What is the correct answer to this question?
A
Why is this the correct answer?
Apply the distributive identity \((X\cup Y)\cap(X\cup Z)=X\cup(Y\cap Z)\). Taking \(X=A\), \(Y=B\), and \(Z=B^c\), the expression becomes \(A\cup(B\cap B^c)\). A set and its complement are disjoint, so \(B\cap B^c=\varnothing\). Therefore the result is \(A\cup\varnothing=A\), making option A correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Venn Diagrams.
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