With respect to a universal set \(U\), which of the following statements is always true for every set \(A\)?
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Answer and explanation
Correct answer: \((A^c)^c=A\)
The complement \(A^c\) contains all elements of the universal set \(U\) that are not in \(A\). Taking the complement once more selects exactly the elements that were originally in \(A\), so \((A^c)^c=A\). In contrast, \(A\cap A^c=\varnothing\) and \(A\cup A^c=U\). Thus, only option A is always true.
Frequently asked questions
What is the correct answer to this question?
\((A^c)^c=A\)
Why is this the correct answer?
The complement \(A^c\) contains all elements of the universal set \(U\) that are not in \(A\). Taking the complement once more selects exactly the elements that were originally in \(A\), so \((A^c)^c=A\). In contrast, \(A\cap A^c=\varnothing\) and \(A\cup A^c=U\). Thus, only option A is always true.
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.
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