For a universal set \(U\), suppose that a set \(A\) is equal to its own complement, \(A=A'\). Which statement about \(U\) must be true?
Advertisement
Answer and explanation
Correct answer: \(U=\varnothing\)
For any set, \(A\cap A'=\varnothing\). If \(A=A'\), then this says \(A\cap A=\varnothing\), so \(A=\varnothing\). Also, \(A\cup A'=U\); replacing \(A'\) by A gives \(A\cup A=A=U\). Thus A must equal both the empty set and U, which is possible only when \(U=\varnothing\).
Frequently asked questions
What is the correct answer to this question?
\(U=\varnothing\)
Why is this the correct answer?
For any set, \(A\cap A'=\varnothing\). If \(A=A'\), then this says \(A\cap A=\varnothing\), so \(A=\varnothing\). Also, \(A\cup A'=U\); replacing \(A'\) by A gives \(A\cup A=A=U\). Thus A must equal both the empty set and U, which is possible only when \(U=\varnothing\).
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.
Advertisement