Which of the following statements is always true for any set A contained in a universal set U?
Answer and explanation
Correct answer: \(A^c\subseteq U\)
By definition, \(A^c=U\setminus A\), meaning that the complement contains only elements taken from the universal set U. Therefore every element of \(A^c\) is necessarily an element of U, so \(A^c\subseteq U\) is always true. The other statements require special conditions: \(A^c=U\) only when A is empty, \(A^c=A\) is not generally true, and \(A^c=\varnothing\) only when A equals U. Hence option A is the universal statement.
Frequently asked questions
What is the correct answer to this question?
\(A^c\subseteq U\)
Why is this the correct answer?
By definition, \(A^c=U\setminus A\), meaning that the complement contains only elements taken from the universal set U. Therefore every element of \(A^c\) is necessarily an element of U, so \(A^c\subseteq U\) is always true. The other statements require special conditions: \(A^c=U\) only when A is empty, \(A^c=A\) is not generally true, and \(A^c=\varnothing\) only when A equals U. Hence option A is the universal statement.
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.
