If \(n(A\cup B)=n(A)+n(B)\) and \(n(A)=0\), what is \(A\cap B\)?
Answer and explanation
Correct answer: \(\varnothing\)
A set with cardinality zero is the empty set, so \(n(A)=0\) implies \(A=\varnothing\). The intersection of the empty set with any set is empty because there is no element that can belong to both sets. Therefore \(A\cap B=\varnothing\). The cardinality equation is also consistent with this: \(n(A\cup B)=n(B)=0+n(B)\).
Frequently asked questions
What is the correct answer to this question?
\(\varnothing\)
Why is this the correct answer?
A set with cardinality zero is the empty set, so \(n(A)=0\) implies \(A=\varnothing\). The intersection of the empty set with any set is empty because there is no element that can belong to both sets. Therefore \(A\cap B=\varnothing\). The cardinality equation is also consistent with this: \(n(A\cup B)=n(B)=0+n(B)\).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
