If the universal set is \(U=\{1,2,3,4,5,6,7,8,9,10\}\), \(A=\{2,4,6,8,10\}\), and \(B=A^c\), what is the value of \(A\cap B\)?
Answer and explanation
Correct answer: \(\varnothing\)
The complement of \(A\) in \(U\) is \(B=\{1,3,5,7,9\}\). The sets \(A\) and \(B\) have no common elements: every element of \(U\) belongs to exactly one of them. Therefore their intersection is empty, \(A\cap B=A\cap A^c=\varnothing\). This is the standard complement identity and not merely a result of the particular numbers used.
Frequently asked questions
What is the correct answer to this question?
\(\varnothing\)
Why is this the correct answer?
The complement of \(A\) in \(U\) is \(B=\{1,3,5,7,9\}\). The sets \(A\) and \(B\) have no common elements: every element of \(U\) belongs to exactly one of them. Therefore their intersection is empty, \(A\cap B=A\cap A^c=\varnothing\). This is the standard complement identity and not merely a result of the particular numbers used.
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.
