The Empty Set, Finite and Infinite Sets, Equal Sets

Correct answer: A. A is infinite

Explanation: There are infinitely many rational numbers strictly between 1 and 2. For example, 3/2, 4/3, and 5/4 are distinct members of A. More generally, rational numbers such as (n + 1)/(n) lie between 1 and 2 for every integer n greater than 1, giving infinitely many different elements. Therefore A is infinite, not a singleton or a finite set.

Correct answer: A. A = B

Explanation: A real number is rational exactly when it can be expressed as p/q with integers p and q, q ≠ 0. The decimal expansion of every rational number terminates or repeats periodically. Conversely, every terminating decimal can be converted to a fraction with a power of 10 as denominator, and every repeating decimal is also rational. Thus A and B contain exactly the same numbers, so A = B.

Correct answer: A. A = B;

Explanation: To determine A, solve x² = 49. The integer solutions are x = 7 and x = −7. The additional condition x < 0 excludes 7 and retains only −7, so A = {−7}. Since B is also defined as the singleton set {−7}, A and B contain exactly the same element. Therefore, A = B. The braces show that each is a set, not merely the number −7.

Correct answer: A. A = ∅

Explanation: Complete the square: x² - 2x + 2 = (x - 1)² + 1. For every real x, (x - 1)² is at least 0, so (x - 1)² + 1 is at least 1 and can never equal 0. Therefore, the equation has no real solution, and the set of real solutions is empty. Hence option A is correct.

Correct answer: A. A = B

Explanation: Rearrange the equation as x² - x - 2 = 0. Factoring gives (x - 2)(x + 1) = 0, so x = 2 or x = -1. Both values are real and therefore A = {-1, 2}. Since B contains exactly these same elements, A = B. The order in which elements are written does not affect a set.

Correct answer: A. Empty set

Explanation: A prime number has exactly two positive factors: 1 and itself. A composite number has more than two positive factors. These definitions are mutually exclusive, so no natural number can be both prime and composite. Since no element satisfies the defining condition, the set contains no elements and is therefore the empty set.

Correct answer: D. A = {0}, B = ∅

Explanation: Set equality depends on the elements present, not their order, repetition, or notation. Thus options A and B describe equal sets: order is irrelevant and repeated 1 is listed only once. Option C also gives two forms of the empty set. In option D, A contains the element 0, whereas B contains no element, so they are not equal.

Correct answer: B. {1, 2} and {3, 4}

Explanation: The sets {1, 2} and {3, 4} each contain two elements, so they have the same cardinality. However, their elements are different, and equality of sets requires exactly the same elements. The other pairs represent equal sets because order does not matter, the two empty-set notations are identical, and repeated elements are ignored. Therefore option B is correct.

Correct answer: B. {x ∈ ℝ : 2 < x < 3}

Explanation: Between any two distinct real numbers, including 2 and 3, there are infinitely many real numbers such as 2.5, 2.25, and 2.125. Therefore {x ∈ ℝ : 2 < x < 3} is an infinite set. Option A contains only 2, option C contains no integer, and option D contains only 2 in ℕ. Hence option B is correct.

Correct answer: D. {x ∈ Z : x² = 16} and {4}

Explanation: Two sets are equal when they contain exactly the same elements, regardless of the way they are described. In option D, the domain is the integers, so x² = 16 gives x = -4 or x = 4. Thus the first set is {-4, 4}, whereas the second set is {4}; because -4 is missing from the second set, the two sets are not equal. The other three pairs contain the same elements.

Correct answer: C. A = {x ∈ N : x < 5}, B = {1, 2, 3, 4}

Explanation: The stem defines N as the positive natural numbers {1, 2, 3, ...}. Therefore, the natural numbers satisfying x < 5 are exactly 1, 2, 3, and 4. Hence A = {1, 2, 3, 4}, which is precisely B in option C. In option A, B incorrectly includes 0; in option B, A also includes 5; and in option D, A contains numbers greater than 5. Thus only C represents equal sets.

Correct answer: C. A = {-√2, √2}, and A is finite

Explanation: Because the domain is the real numbers, both real square roots of 2 must be considered. Solving x² = 2 gives x = √2 or x = -√2. Therefore A = {-√2, √2}. These are two distinct real numbers, so the set has exactly two elements and is finite. A common error is to write only √2 and overlook the negative solution, even though both numbers have square 2.

Correct answer: B. {x ∈ N : x is prime and x < 1000}

Explanation: There are only finitely many natural numbers less than 1000, namely the numbers from 1 through 999. The prime numbers satisfying the condition form a subset of this finite collection, so their number is also finite, even though the list may be fairly long. In contrast, there are infinitely many natural primes, infinitely many negative integers, and infinitely many real numbers between 0 and 1. Therefore option B is the finite set.

Correct answer: B. A = {1}, so A ≠ B

Explanation: The equation x² − 1 = 0 factors as (x − 1)(x + 1) = 0, giving x = 1 or x = −1. However, A is restricted to the natural numbers. Under the usual convention N = {1, 2, 3, ...}, only 1 is admitted, so A = {1}. Set B contains both −1 and 1, and therefore A ≠ B.

Correct answer: A. A = B, because both sets contain only the element 3

Explanation: The quadratic x² − 6x + 9 is (x − 3)², so its only integer solution is x = 3. Thus A = {3}. In set notation, repetitions do not create additional elements; writing 3 three times still represents B = {3}. Consequently A and B contain exactly the same element and are equal. Repeated roots affect multiplicity in algebra, but not the number of distinct elements in a set.

Correct answer: C. It is an infinite set

Explanation: There are infinitely many rational numbers strictly between 0 and 1. For example, 1/2, 1/3, and 2/3 belong to A, and for every positive integer n, the rational number 1/(n + 1) also lies between 0 and 1. Thus new distinct elements can be generated endlessly. The endpoints 0 and 1 are excluded, but excluding them does not make the set finite.

Correct answer: A. Empty set

Explanation: The expression |x − 2| + |x − 5| represents the sum of the distances from x to 2 and 5 on the real number line. By the triangle inequality, this sum is always at least the distance between 2 and 5, which is 3. Since the question requires the sum to equal 2, no real number x can satisfy it. Hence A is the empty set.

Correct answer: A. {x² : x ∈ N, 1 ≤ x ≤ 4} = {1, 4, 9, 16}

Explanation: The natural numbers satisfying 1 ≤ x ≤ 4 are 1, 2, 3, and 4. Squaring these values gives 1² = 1, 2² = 4, 3² = 9, and 4² = 16. Therefore, the set described by {x² : x ∈ N, 1 ≤ x ≤ 4} contains exactly the four elements {1, 4, 9, 16}, so option A is equal to the given set. The other options either contain unsquared numbers, include an extra element such as 0, or contain many more elements.

Correct answer: B. A = {x ∈ Z : x² = 1}, B = {1}

Explanation: For option B, solving x² = 1 over the integers gives x = −1 or x = 1. Thus A = {−1, 1}, which has two elements, while B = {1}, which has one element. Both sets are finite, but they are not equal because −1 belongs to A and does not belong to B. Option A describes equal sets, option C has an infinite first set, and option D has two equal empty sets.

Correct answer: A. A = B

Explanation: Factor the quadratic expression as x² − 7x + 12 = (x − 3)(x − 4). Therefore the equation is satisfied when x = 3 or x = 4. Both values are integers, so A = {3, 4}. Since B is also {3, 4}, the two sets contain exactly the same elements and are equal. The numbers 7 and 12 are coefficients and are not the solutions themselves. Thus option A is correct.

Correct answer: B. Empty set

Explanation: The condition 2x + 1 = 2x can be simplified by subtracting 2x from both sides, giving 1 = 0. This statement is impossible and is independent of the value of x. Therefore no natural number satisfies the defining condition of A. A set containing no elements is called the empty set, denoted by ∅. Hence option B is correct; it is not a singleton, a two-element set, or an infinite set.

Correct answer: C. Infinite set

Explanation: The congruence x ≡ 2 (mod 5) means that x has the form x = 5k + 2, where k is any integer. Taking k = 0, 1, 2 gives 2, 7, 12, while k = −1, −2 gives −3, −8. Since there are infinitely many possible integer values of k, there are infinitely many elements in A. Thus option C is correct.

Correct answer: B. A = ∅

Explanation: Complete the square: x² + 2x + 2 = (x + 1)² + 1. For every real x, (x + 1)² is at least zero, so (x + 1)² + 1 is always at least one and can never equal zero. Equivalently, the discriminant is 2² − 4(1)(2) = −4, which is negative. Hence there is no real solution and A is the empty set.

Correct answer: B. {−2}

Explanation: Use the standard absolute-value inequality: |x + 2| < 1 is equivalent to −1 < x + 2 < 1. Subtracting 2 throughout gives −3 < x < −1. Among integers, the only number strictly between −3 and −1 is −2. The endpoints −3 and −1 are excluded because the inequality is strict. Thus A contains exactly one element and A = {−2}.

Correct answer: B. {x ∈ ℕ : x > 100} and {x ∈ ℤ : x < 0}

Explanation: The natural numbers greater than 100 are 101, 102, 103, and so on without end, so the first set in option B is infinite. The negative integers less than zero are −1, −2, −3, and so on, which also continue indefinitely; therefore the second set is infinite. The other options contain bounded sets, divisors of 100, or sets with only zero or one element.