Let U = {x ∈ ℤ | 1 ≤ x ≤ 30} and A = {x ∈ U | x ≡ 1 (mod 4)}. What is n(A′), the number of elements in the complement of A with respect to U?
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Answer and explanation
Correct answer: 22
The universal set U contains every integer from 1 through 30, so n(U) = 30. The integers congruent to 1 modulo 4 in this interval are 1, 5, 9, 13, 17, 21, 25, and 29; hence n(A) = 8. Since A′ contains all elements of U not in A, n(A′) = n(U) − n(A) = 30 − 8 = 22. Therefore option A is correct.
Frequently asked questions
What is the correct answer to this question?
22
Why is this the correct answer?
The universal set U contains every integer from 1 through 30, so n(U) = 30. The integers congruent to 1 modulo 4 in this interval are 1, 5, 9, 13, 17, 21, 25, and 29; hence n(A) = 8. Since A′ contains all elements of U not in A, n(A′) = n(U) − n(A) = 30 − 8 = 22. Therefore option A is correct.
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.
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