Sets and their representations

Correct answer: B. 8

Explanation: The condition requires 1 to be included and 2 to be excluded, so both of these elements have fixed choices. The remaining elements 3, 4, and 5 are unrestricted; each can either be included or left out independently. Therefore, the number of possible subsets is \(2\times2\times2=2^3=8\). For example, \{1\}, \{1,3\}, and \{1,4,5\} are valid, while any subset containing 2 is invalid. Thus option B is correct.

Correct answer: A. A = B

Explanation: The positive divisors of 18 are 1, 2, 3, 6, 9, and 18. Applying the condition “less than 10” removes 18, so A = {1, 2, 3, 6, 9}. This is exactly the same collection of elements as B. In a set, order does not matter, but every element must match. Therefore A = B, making option A correct. Option D incorrectly includes 18, which does not satisfy the given inequality.

Correct answer: A. A = B

Explanation: For an absolute-value inequality, |x − 2| < 2 is equivalent to −2 < x − 2 < 2. Adding 2 throughout gives 0 < x < 4. Since x must be an integer, the only possible values are 1, 2, and 3. Thus A = {1, 2, 3}, which is exactly B. Therefore A = B is correct; neither set is a proper subset of the other, and their intersection is not empty.

Correct answer: C. 8

Explanation: Consider the pair 2 and 3. There are two allowed cases: both are included, or both are excluded. In either case, the remaining elements 1 and 4 can each be independently included or excluded, giving 2^2 = 4 choices. Thus the total number is 4 + 4 = 8, or equivalently 2 × 2^2 = 8. Hence option C is correct.

Correct answer: C. 23

Explanation: Use the inclusion–exclusion formula first: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Hence 50 = 35 + 27 − n(A ∩ B), so n(A ∩ B) = 12. The set A is made up of the disjoint parts A − B and A ∩ B. Therefore n(A − B) = n(A) − n(A ∩ B) = 35 − 12 = 23. Option B is the intersection size, not the difference size; the other values do not satisfy the given relationships.

Correct answer: C. 4

Explanation: For every \(x\in A\), calculate \(y=2x\) and check whether the result belongs to \(B\). The values are \(2,4,8,16\) for \(x=1,2,4,8\), respectively. Therefore, the valid ordered pairs are \((1,2),(2,4),(4,8),(8,16)\). All four satisfy the condition and belong to \(A\times B\), so the answer is 4, option C.

Correct answer: C. 10

Explanation: Fix each value of \(x\) and count the allowed values of \(y\). For \(x=1\), all four values of \(y\) work. For \(x=2,3,4\), the numbers of choices are 3, 2, and 1, respectively. Hence the total is \(4+3+2+1=10\). Because ordered pairs are being counted, each distinct position of \(x\) and \(y\) is included, so option C is correct.