The Empty Set, Finite and Infinite Sets, Equal Sets

The governing concept is equality of sets. In a set, the order of elements has no significance, and repeating an element does not create an additional distinct member. Therefore B simplifies to {a, b, c, d}, exactly the same collection as A. The sets are equal, so option A is correct. Option B wrongly treats order or repetition as important, while C and D misclassify a set with four distinct elements.

The equation x + 5 = 5 gives x = 0. In this question, the usual school convention ℕ = {1, 2, 3, ...} is being used, so 0 is not a natural number. Consequently, no natural number satisfies the condition, and A has no elements. Therefore A is the empty set, written as ∅.

The condition includes every integer from −3 through 3, including both endpoints. Therefore, B = {−3, −2, −1, 0, 1, 2, 3}. Counting these distinct integers gives 7 elements. Since the interval is bounded and contains only finitely many integers, B is a finite set. Hence option B is correct.

The governing concept is equality of sets after expanding a set-builder description. The natural even numbers less than 10 are 2, 4, 6, and 8, so B = {2, 4, 6, 8}. This is exactly the element collection shown for A; set equality does not depend on the order of writing. Thus option A is correct. B is not empty, and A has only four elements, so it is not infinite.

In a set, repeating an element does not create a new element. Hence C can be written as {1, 2, 3}. Set D contains the same three elements, written in a different order: {3, 2, 1}. Since set equality depends on having exactly the same elements, not on order or repetition, C = D. Therefore, option A is correct.

An empty set is a set that contains no elements. Its cardinality, or number of elements, is zero. It is commonly represented by the symbols ∅ or { }. This is different from a singleton set, which contains exactly one element, and from finite or infinite non-empty sets. Therefore, option A gives the correct definition.

Under the stated convention, the natural numbers begin with 1. Every natural number is therefore at least 1, so none of them can satisfy x < 1. Since the defining condition has no solution in N, the set contains no elements and is the empty set, written as ∅. It is finite with cardinality zero.

A calendar month may have 28, 29, 30, or 31 days, but no month has 32 days. Therefore no month satisfies the stated property. A set whose defining condition is satisfied by no object is the empty set. It is finite as well, with cardinality zero, but the most specific answer here is empty set.

A finite set has a fixed and limited number of distinct elements. Therefore, its elements can be listed or counted completely, even if the number is large. For example, {2, 4, 6, 8} has four elements. The empty set is also finite because it has zero elements. A set does not need to contain natural numbers to be finite.

The set contains all positive even numbers from 2 through 50. These can be written as 2 × 1, 2 × 2, ..., 2 × 25, so each integer from 1 to 25 gives exactly one element. Alternatively, for an arithmetic sequence, the number of terms is (last term − first term) ÷ common difference + 1 = (50 − 2) ÷ 2 + 1 = 25. Hence the set has 25 elements.

The integers strictly between 20 and 30 are 21 through 29. Among them, 23 and 29 are prime: each has exactly two positive divisors, 1 and itself. The numbers 21 and 27 are composite, so they cannot be included. Thus A = {23, 29}, which has two elements and is therefore finite.

The set is {101, 102, 103, ...}. After every listed positive integer, another larger positive integer exists, so the sequence never reaches a final element. Consequently, the elements cannot be completely counted or listed in a finite list. The set therefore has infinitely many elements and is an infinite set.

The multiples of 7 in the natural numbers are 7, 14, 21, 28, 35, and so on. For every natural number n, 7n is another multiple of 7, so the list never ends. Therefore, the set has infinitely many elements and is an infinite set. It is not empty, a singleton, or a set containing only seven elements.

Whole numbers begin with 0 and continue as 1, 2, 3, and so on. The whole numbers less than 10 are exactly {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. This is a complete list containing ten distinct elements, so the set is finite. It is not empty, because it contains elements, and it is not infinite because the list ends at 9.

A straight line is continuous and contains points between every pair of its points. No matter how many points are identified, more points can always be found on the same line. Therefore the collection cannot be completely listed as a finite set. The set of all points on a straight line is infinite.

Through one fixed point, a line can be drawn in every possible direction. Different directions produce different lines, and there is no finite limit on the number of directions or lines. Hence the collection contains infinitely many lines. It is therefore an infinite set, not an empty, finite, or two-element set.

The only even prime number is 2. Every other even number is divisible by 2 and therefore has at least two positive divisors, so it is not prime. Since the question asks for even prime numbers greater than 2, no number satisfies the condition. Hence the set has no elements and is the empty set, written as ∅.

The notation {0} means a set whose single element is the number 0. It is important not to confuse the set {0} with the empty set ∅. The empty set contains no elements, whereas {0} contains exactly one element. Therefore, {0} is a non-empty singleton set and option A is correct.

The condition x² = 9 means that x can be 3 or −3, because both (3)² and (−3)² equal 9. Both values belong to the set of integers ℤ, so both must be included in A. Therefore A = {−3, 3}. The values −9 and 9 are not solutions because the equation involves the square of x.

The positive divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. This list contains eight distinct numbers and is complete because no other positive integer divides 24 without a remainder. Since the set can be listed completely and has a fixed number of elements, it is finite. Therefore, option A is correct.

For any number x, the equation x + 3 = x would require subtracting x from both sides, giving 3 = 0. This is impossible, so no natural number satisfies the stated condition. A set defined by a condition has no elements when the condition has no solution. Therefore, the given solution set is the empty set ∅.

A real number has square zero only when the number itself is zero. Thus the equation x² = 0 has the single solution x = 0, and 0 belongs to ℝ. The set is therefore {0}, which contains exactly one element. A one-element set is called a singleton and is finite, so option A is correct.

There are infinitely many rational numbers strictly between 0 and 1. Examples include 1/2, 1/3, 2/3, and 1/n for every integer n greater than 1. More generally, between any two distinct rational numbers there are further rational numbers. Therefore, the set of rational numbers in the interval (0, 1) is infinite.

The integers are ..., −2, −1, 0, 1, 2, ... . The numbers 0 and 1 are consecutive integers, so there is no integer strictly between them. Because the interval is strict, 0 and 1 themselves are not included either. Consequently, no integer satisfies the condition, and the set is empty: ∅.

The interval (0, 1) contains infinitely many real numbers. It includes rational numbers such as 1/2 and 3/4, as well as irrational numbers such as √2/2. In fact, between any two distinct real numbers there are infinitely many other real numbers. Thus the set of real numbers strictly between 0 and 1 is infinite.