If \(A\cup B=U\), what is the value of \(A'\cap B'\)?
Answer and explanation
Correct answer: \(\varnothing\)
By De Morgan’s law, the intersection of the complements equals the complement of the union: \(A'\cap B'=(A\cup B)'\). The question states that \(A\cup B=U\), where \(U\) is the universal set. The complement of the universal set contains no elements, so \(U'=\varnothing\). Hence \(A'\cap B'=\varnothing\), and option C is correct. Option D is a different expression: by De Morgan’s law, \(A'\cup B'=(A\cap B)'\), so it cannot be selected.
Frequently asked questions
What is the correct answer to this question?
\(\varnothing\)
Why is this the correct answer?
By De Morgan’s law, the intersection of the complements equals the complement of the union: \(A'\cap B'=(A\cup B)'\). The question states that \(A\cup B=U\), where \(U\) is the universal set. The complement of the universal set contains no elements, so \(U'=\varnothing\). Hence \(A'\cap B'=\varnothing\), and option C is correct. Option D is a different expression: by De Morgan’s law, \(A'\cup B'=(A\cap B)'\), so it cannot be selected.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Complement of a Set and Its Properties.
