If the universal set is \(U=\{1,2,3,\ldots,12\}\), \(A=\{3,6,9,12\}\), and \(B=A^c\), what is the value of \(A\cap B\)?
Answer and explanation
Correct answer: \(\varnothing\)
The complement \(B=A^c\) contains exactly those elements of the universal set \(U\) that are not members of \(A\). Therefore, no element can belong to both \(A\) and \(B\) at the same time. Hence, \(A\cap B=A\cap A^c=\varnothing\). The correct answer is option A. This is a standard complement identity, and it is true for every set relative to its universal set.
Frequently asked questions
What is the correct answer to this question?
\(\varnothing\)
Why is this the correct answer?
The complement \(B=A^c\) contains exactly those elements of the universal set \(U\) that are not members of \(A\). Therefore, no element can belong to both \(A\) and \(B\) at the same time. Hence, \(A\cap B=A\cap A^c=\varnothing\). The correct answer is option A. This is a standard complement identity, and it is true for every set relative to its universal set.
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.
