If n(U) = 170, n(A ∩ B) = 44, and n(Aᶜ ∪ Bᶜ) = 146, what can be said about the given data?
Answer and explanation
Correct answer: The data are inconsistent
By De Morgan’s law, \(A^c\cup B^c=(A\cap B)^c\). Therefore its cardinality must be \(n(U)-n(A\cap B)=170-44=126\). The stated value is 146, which differs from 126 by 20. Since both values cannot describe the same complement in the same universal set, the supplied data are inconsistent. Hence option A is correct.
Frequently asked questions
What is the correct answer to this question?
The data are inconsistent
Why is this the correct answer?
By De Morgan’s law, \(A^c\cup B^c=(A\cap B)^c\). Therefore its cardinality must be \(n(U)-n(A\cap B)=170-44=126\). The stated value is 146, which differs from 126 by 20. Since both values cannot describe the same complement in the same universal set, the supplied data are inconsistent. Hence option A is correct.
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.
