If A is a proper subset of the universal set U, which statement about A^c is true?
Answer and explanation
Correct answer: \(A^c\ne\varnothing\)
A proper subset A of U is contained in U but is not equal to U. Thus, at least one element of U is not contained in A. By definition, all such elements form the complement A^c, so A^c must contain at least one element and cannot be empty. Also, every complement is a subset of U, which rules out option D. Therefore, option A is the only correct statement.
Frequently asked questions
What is the correct answer to this question?
\(A^c\ne\varnothing\)
Why is this the correct answer?
A proper subset A of U is contained in U but is not equal to U. Thus, at least one element of U is not contained in A. By definition, all such elements form the complement A^c, so A^c must contain at least one element and cannot be empty. Also, every complement is a subset of U, which rules out option D. Therefore, option A is the only correct 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.
