Complement of a Set and Its Properties

The complement of A, written as A′, contains every element of the universal set U that is not present in A. From U = {0,1,2,3,4,5} and A = {0,1,5}, we get A′ = {2,3,4}. This set has three elements, so n(A′) = 3 and option B is correct. Equivalently, n(A′) = n(U) − n(A) = 6 − 3 = 3. Option D gives the size of U, not its complement.

The complement of G is taken relative to the universal set U. Therefore, G′ contains every student in the class who is not in G, meaning every student who does not wear glasses. It does not represent teachers or only absent students, because those groups are not defined by the given universal set and condition.

For every element of the universal set U, exactly one of two possibilities holds: it is in A, or it is not in A and therefore belongs to A′. Consequently, combining A and A′ includes every element of U and no element outside U. Hence, A ∪ A′ = U. This is one of the fundamental complement identities.

The complement A′ contains elements of U that are not in A. Therefore, no element can belong to both A and A′ at the same time. Their common part, or intersection, contains no elements and is the empty set. Hence, A ∩ A′ = ∅. This identity is paired with A ∪ A′ = U and applies for complements taken relative to the same universal set U.

The complement A′ contains exactly those elements of the universal set U that are not members of A. Starting with U = {1, 2, 3, 4, 5, 6, 8}, remove 2, 4, 6, and 8, because these belong to A. The elements left are 1, 3, and 5. Hence A′ = U − A = {1, 3, 5}.

The complement A′ and the set A together divide the universal set U into two non-overlapping parts. Therefore, A is obtained by removing the elements of A′ from U: A = U − A′ = {1, 2, 3, 4} − {1, 4} = {2, 3}. Option A gives A′ itself, not A. The answer follows from the complement relationship A ∪ A′ = U.

The complement A′ is defined relative to U and consists of every element of U that is not in A. From U = {sun, moon, star}, remove sun because it is the only element of A. The remaining elements are moon and star, so A′ = U − A = {moon, star}. Thus option B is correct; the other options omit a required element or repeat A.

The complement A′ contains those elements of the universal set U that are not members of A. Starting with U = {1,2,3,4,5,6,7,8} and removing A = {1,2,3,4}, we obtain A′ = {5,6,7,8}. The smallest element of this remaining set is 5, so option C is correct. The number 4 is excluded because it belongs to A.

The complement of A in U is A' = U \ A, meaning the elements of U that are not in A. Here A and U are exactly the same set, so there are no elements of U left after removing A. Therefore A' = ∅, and the empty set has zero elements. Hence the correct answer is 0.

The complement A′ in U consists of all elements of U that are not in A. From U = {black, white, red, yellow}, remove the elements black and yellow because they belong to A. The remaining elements are white and red, so A′ = {white, red}. Therefore option B is correct; a complement cannot contain any element already present in A.

The complement M' contains all elements of the universal set U that are not members of M. Removing 2, 4, 6, 8, and 10 from U gives M' = {1, 3, 5, 7, 9}. Among the choices, only 9 appears in this complement. Therefore, option C is correct; 2, 6, and 10 belong to M itself.

The complement is taken with respect to U. Removing the elements of A from U gives A' = U − A = {1, 4, 6}. Among these elements, only 1 is strictly less than 4; the number 4 itself is not less than 4. Therefore, exactly one element satisfies the condition, so option A is correct.

The complement of A in U is A' = U − A = {2, 3, 4}. Taking the complement once more gives (A')' = U − {2, 3, 4} = {1, 5}, which is the original set A. This illustrates the double-complement law: the complement of a complement returns the original set, provided the same universal set is used. Hence option A is correct.

The complement A' consists of all elements in the universal set U that are not in A. The digits are 0 through 9, while A contains all even digits: 0, 2, 4, 6, and 8. Removing them from U leaves the odd digits 1, 3, 5, 7, and 9. Therefore A' is {1, 3, 5, 7, 9}, so option A is correct.

The complement B = A' is taken with respect to the universal set U. Since A = {1,2,3}, its complement is B = {4,5,6}. Taking the complement once more gives B' = (A')' = A, by the double-complement law. Therefore B' = {1,2,3}. Option B is B itself, not B', while option C is the universal set and option D is the empty set.

The complement A' contains every element of the universal set U that is not an element of A. Starting with U = {2,4,6,8,10,12}, remove 4, 8, and 12 because they belong to A. The elements left are 2, 6, and 10. Therefore A' = {2,6,10}. Option B is A itself, and the other options omit or incorrectly include elements.

The complement of A relative to U is defined as A' = U \ A: all elements in U that are not in A. From U = {1,2,3,4,5,6,7,8}, remove 2, 4, and 8. The remaining elements are {1,3,5,6,7}, so option B is correct. Option A repeats A itself, while option C still contains elements of A and option D is not justified.

The complement C' contains elements of the universal set U that are not contained in C. The elements a, c, and f are already in C, so they cannot be in C'. Removing them from U = {a,b,c,d,f} leaves C' = {b,d}. Therefore b is in C'. Although d is also in the complement, it is not among the answer choices; among the given options, only b is correct.

The complement \(A'\) contains all elements of the universal set \(U\) that are not elements of \(A\). Removing 1, 3, and 5 from \(U\) gives \(A'=\{0,2,4,6\}\). This set has four elements, so \(n(A')=4\). The same result follows from the formula \(n(A')=n(U)-n(A)=7-3=4\). Thus option B is correct; 3 is the size of \(A\), while 7 is the size of \(U\).

The complement of a set is always defined relative to its universal set. Here, \(U\) contains every book in the library, while \(H\) contains only the Hindi books. Therefore, \(H'=U\setminus H\) consists of all books in the library that are not Hindi books, such as English or other-language books. It does not refer to authors, publication age, or books outside the library. Hence option B is correct.