The Empty Set, Finite and Infinite Sets, Equal Sets

Correct answer: A. ∅ ⊆ A

Explanation: The empty set has no elements. The statement ∅ ⊆ A means that every element of ∅ is also an element of A. Because there are no elements in ∅, there is no counterexample to this requirement; therefore the statement is true for every set A. In contrast, A ⊆ ∅ is true only when A itself is empty, and ∅ = A is not always true. Hence option A is correct.

Correct answer: A. They are equal sets

Explanation: The order of elements does not matter in a set. Two sets are equal when they contain exactly the same elements. Both given sets contain 1, 2, and 3, even though the elements are written in different orders. Therefore, {1, 2, 3} = {3, 2, 1}; they are equal sets, and option A is correct.

Correct answer: A. A and B are equal sets

Explanation: The positive natural-number divisors of 9 are 1, 3, and 9. Therefore the set described by A is \(A=\{1,3,9\}\), which is exactly the given set B. Since two sets are equal when they contain precisely the same elements, A and B are equal. They are not disjoint, empty, or in a proper-subset relationship. The order in which elements are listed would not affect equality, although here the order is already the same.

Correct answer: A. A = B

Explanation: Using the standard school convention N = {1, 2, 3, ...}, the natural numbers not exceeding 4 are 1, 2, 3, and 4. Hence A = {1, 2, 3, 4}. Set B contains exactly these same elements, merely written in reverse order. The order of elements does not matter in a set, so A = B. Option B is false for this reason; option C is false because equal sets are not proper subsets of one another; and option D is false because B has four elements.

Correct answer: A. A = B

Explanation: Prime divisors are prime numbers that divide the given number exactly. The prime factorization of 42 is 42 = 2 × 3 × 7. Therefore, the complete set of prime divisors of 42 is A = {2, 3, 7}. This is exactly the set specified as B, so A = B. Options B and C are incorrect because equal sets are not proper subsets of one another. Option D is false because A contains three elements and is therefore not empty.

Correct answer: A. A = B

Explanation: To expand B, list the positive even integers less than 8. They are 2, 4, and 6; 8 is not allowed because the inequality is strict, and neither 0 nor negative even numbers are positive. Therefore B = {2,4,6}, which has exactly the same elements as A. Hence A = B. Options B and C incorrectly claim a proper-subset relation, while option D is false because A ∩ B = {2,4,6}, not the empty set.

Correct answer: A. A and B are equal

Explanation: The positive multiples of 6 that are less than 20 are 6, 12, and 18. The next positive multiple is 24, which is not less than 20. Thus A = {6,12,18}. This is exactly the set B, so A and B are equal. The set is finite because it has only three elements, and neither set is a proper subset of the other. Therefore, option A is correct.

Correct answer: B. {2, 4, 6}

Explanation: An even integer is divisible by 2 without leaving a remainder. Checking the elements of A one by one, 2, 4, and 6 are even, whereas 1, 3, and 5 are odd. Therefore the set containing all even elements of A is B = {2, 4, 6}. Option C misses 6, option A contains the odd elements, and option D includes every element rather than only the even ones.

Correct answer: A. A = B

Explanation: The positive divisors of 15 are the positive numbers that divide 15 exactly: 1, 3, 5, and 15. Thus the rule defining A gives A = {1, 3, 5, 15}. This is precisely the list used to define B, so A and B contain the same elements and are equal. Options B and C incorrectly claim a proper-subset relation, while D omits 1 and 15.

Correct answer: A. ∅ ⊆ A

Explanation: The empty set ∅ is a subset of every set A. A subset relation requires every element of the first set to belong to the second set. Since ∅ has no elements, there is no element that can violate this condition; consequently, ∅ ⊆ A is always true. The other statements require additional conditions: A must be empty for A ⊆ ∅ or ∅ = A, and ∅ ∈ A depends on whether A contains the empty set as an element.

Correct answer: A. No

Explanation: Having the same cardinality means only that the two finite sets contain the same number of elements; it does not mean that the elements themselves are identical. For example, \(A=\{1,2\}\) and \(B=\{3,4\}\) both have cardinality 2, but they are different sets. Equality requires every element of A to be in B and every element of B to be in A.

Correct answer: B. 3

Explanation: Two singleton sets are equal precisely when their only elements are equal. Therefore, from {2a + 1} = {7}, we obtain the equation 2a + 1 = 7. Subtracting 1 from both sides gives 2a = 6, and dividing by 2 gives a = 3. Substitution verifies the result: 2(3) + 1 = 7, so both singleton sets become {7}. Thus option B is the unique correct answer.

Correct answer: A. \(\{-2,-1,0,1,2\}\)

Explanation: The notation says that \(x\) must be an integer greater than \(-3\) and less than or equal to \(2\). The integers greater than \(-3\) begin with \(-2\), and the upper endpoint \(2\) is included because of the symbol \(\le\). Thus the complete list is \(\{-2,-1,0,1,2\}\), which is option A. Option B incorrectly includes \(-3\); option C omits \(2\); and option D includes \(-3\) while omitting \(2\).

Correct answer: C. 7

Explanation: Each side is a singleton set, meaning that each set has exactly one element. Two singleton sets are equal only when their sole elements are equal, so \(a+2=9\). Subtracting 2 from both sides gives \(a=7\), which is option C. The braces indicate sets; they do not change the equality principle. A common error is to add 2 to 9 and choose 11, but the equation requires subtraction because 2 is already added to \(a\).

Correct answer: A. A is non-empty

Explanation: The statement ∅ ∈ A says that the empty set itself is an element of A. Therefore, A contains at least one element, namely ∅, and so A is non-empty. It does not mean that A equals the empty set; in fact, the empty set has no elements. Also, every set has at least the empty subset, so the remaining statements are not valid.

Correct answer: A. A = B

Explanation: To list B, substitute the allowed values n = 1, 2, 3, and 4 into x = 2n. This gives x = 2, 4, 6, and 8, so B = {2, 4, 6, 8}. This roster is exactly the same as A, meaning the two sets have identical elements and therefore A = B. Neither set is a proper subset of the other, because proper inclusion requires unequal sets. Their intersection is also the whole set, not empty.

Correct answer: B. 2

Explanation: The set A has exactly two direct elements: the set {1} and the number 2. Although 1 appears inside the element {1}, it is not separately listed as an element of A. Therefore, the number of elements or cardinality of A is n(A) = 2. Nested elements must not be counted again.

Correct answer: B. 3

Explanation: Cardinality counts the elements at the outermost level of a set. In \(A=\{\{2\},4,6\}\), the first element is the set \(\{2\}\) itself, the second element is 4, and the third element is 6. The number 2 is inside the first element and is not a separate outer element of \(A\). Therefore \(A\) has three elements and \(n(A)=3\), making option B correct.

Correct answer: B. U

Explanation: The complement A' is defined as the set of all elements in the universal set U that are not in A. If A is the empty set, it contains no elements, so no element of U is removed. Consequently every element of U belongs to A', and therefore ∅' = U. Option A confuses a set with its complement, while P(U) is a different power set.

Correct answer: A. A ⊆ B

Explanation: A−B is the portion of A lying outside B. If its cardinality is zero, that portion contains no elements, so no element of A can lie outside B. Therefore every element of A belongs to B, which is exactly the definition of A⊆B. The condition n(A)>0 only confirms that A is nonempty; it does not imply B⊆A or disjointness.

Correct answer: A. 109

Explanation: The union contains all elements of both sets, so it must contain at least as many elements as the larger set. Therefore n(A ∪ B) ≥ max(n(A), n(B)) = 109. This lower bound is attainable if the smaller set B is completely contained in A. In that case A ∪ B = A and its cardinality is 109. Hence the minimum possible value is 109, option A.

Correct answer: A. \(\{1,2,3\}\)

Explanation: The intersection consists of all elements that belong to both sets. The order is irrelevant in set notation, so \(B=\{3,2,1\}\) is the same set as \(\{1,2,3\}\). Every element 1, 2, and 3 is common to A and B. Therefore, \(A\cap B=\{1,2,3\}\), making option A correct.

Correct answer: A. 6

Explanation: The union of two sets contains every distinct element appearing in either set, but a common element is counted only once. Here, \(A\cup B=\{1,2,3,4,5,6\}\), so its cardinality is 6. Equivalently, \(n(A\cup B)=4+3-1=6\), because 4 belongs to both sets. Option B incorrectly counts 4 twice; options C and D do not represent the complete union.

Correct answer: B. 3

Explanation: The equation \(x+y=0\) is equivalent to \(y=-x\). We must ensure that the first component belongs to A and the second belongs to B. For \(x=-1\), \(y=1\); for \(x=0\), \(y=0\); and for \(x=1\), \(y=-1\). These give \((-1,1),(0,0),(1,-1)\). Values \(x=-2\) and \(x=2\) would require 2 and -2, neither of which is in B. Therefore, the answer is 3, option B.