If \(U=\{1,2,3,\ldots,18\}\), \(A=\{x:x\text{ is divisible by }2\}\), and \(B=\{x:x\text{ is divisible by }3\}\), how many elements are in \(A^c\cap B^c\)?
Answer and explanation
Correct answer: 6
The set \(A^c\cap B^c\) contains numbers in U that are divisible by neither 2 nor 3. Listing the integers from 1 to 18 and removing all even numbers and all multiples of 3 leaves \(\{1,5,7,11,13,17\}\). This set has six elements. Equivalently, De Morgan’s law gives \(A^c\cap B^c=(A\cup B)^c\), which describes numbers divisible by neither divisor.
Frequently asked questions
What is the correct answer to this question?
6
Why is this the correct answer?
The set \(A^c\cap B^c\) contains numbers in U that are divisible by neither 2 nor 3. Listing the integers from 1 to 18 and removing all even numbers and all multiples of 3 leaves \(\{1,5,7,11,13,17\}\). This set has six elements. Equivalently, De Morgan’s law gives \(A^c\cap B^c=(A\cup B)^c\), which describes numbers divisible by neither divisor.
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.
