Let \(U\) be a universal set with \(n(U)=200\). If \(n(A\cup B)=128\), \(n(A\cap B)=36\), and \(n(A^c\cap B^c)=72\), which relation is true?
Advertisement
Answer and explanation
Correct answer: \(n(A\cup B)+n(A^c\cap B^c)=n(U)\)
By De Morgan’s law, \((A\cup B)^c=A^c\cap B^c\). Thus, \(A\cup B\) and \(A^c\cap B^c\) are complementary regions of the universal set and their cardinalities add to \(n(U)\). Numerically, \(128+72=200\), so option A is true. The intersection count 36 is not complementary to the union.
Frequently asked questions
What is the correct answer to this question?
\(n(A\cup B)+n(A^c\cap B^c)=n(U)\)
Why is this the correct answer?
By De Morgan’s law, \((A\cup B)^c=A^c\cap B^c\). Thus, \(A\cup B\) and \(A^c\cap B^c\) are complementary regions of the universal set and their cardinalities add to \(n(U)\). Numerically, \(128+72=200\), so option A is true. The intersection count 36 is not complementary to the union.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Venn Diagrams.
Advertisement