If \(A=\{x\in\mathbb{N}:x\le 15\}\), \(B=\{x\in\mathbb{N}:3\mid x\}\), and \(C=\{x\in\mathbb{N}:5\mid x\}\), where \(\mathbb{N}=\{1,2,3,\ldots\}\), what is \(A\cap(B\cup C)\)?
Answer and explanation
Correct answer: \(\{3,5,6,9,10,12,15\}\)
Within the restriction \(x\le15\), the multiples of 3 are \(\{3,6,9,12,15\}\), and the multiples of 5 are \(\{5,10,15\}\). Their union contains every number divisible by 3 or 5: \(\{3,5,6,9,10,12,15\}\). Intersecting with \(A\) simply enforces the upper bound of 15. Thus option A is correct; options C and D omit one of the two groups.
Frequently asked questions
What is the correct answer to this question?
\(\{3,5,6,9,10,12,15\}\)
Why is this the correct answer?
Within the restriction \(x\le15\), the multiples of 3 are \(\{3,6,9,12,15\}\), and the multiples of 5 are \(\{5,10,15\}\). Their union contains every number divisible by 3 or 5: \(\{3,5,6,9,10,12,15\}\). Intersecting with \(A\) simply enforces the upper bound of 15. Thus option A is correct; options C and D omit one of the two groups.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
