The Empty Set, Finite and Infinite Sets, Equal Sets

A number must satisfy both conditions to belong to A: it must be even and prime. The only even prime number is 2, because every other even natural number greater than 2 is divisible by 2 and is therefore composite. Thus A contains exactly one element and is the singleton set {2}. Option B is incorrect because the set is not empty; C includes composite numbers, and D includes 1, which is not prime.

A prime number has exactly two distinct positive factors, whereas a composite natural number has more than two positive factors. These definitions are mutually exclusive, so no natural number can be prime and composite at the same time. Consequently, the set contains no elements and is the empty set, written as ∅.

For option A, assuming N = {1, 2, 3, ...}, the condition x ≤ 3 gives the elements 1, 2, and 3. Thus the first set is exactly {1, 2, 3}, which is the second set. In the other options, one element is missing or an extra element is present. Equality of sets depends on having precisely the same elements, not on their order.

The empty set is defined as the set that contains no elements. The number of elements of a set is called its cardinality, so the cardinality of the empty set is n(∅) = 0. It is important not to confuse ∅ with {0}: the former has no elements, while the latter contains the element 0 and therefore has one element.

The notation {∅} means a set whose single element is the empty set itself. Therefore its cardinality is 1. This differs from ∅, which has no elements and cardinality 0. Braces are significant: ∅ denotes nothing inside the set, whereas {∅} contains one object, namely the empty set. Thus n({∅}) = 1.

Equal sets contain exactly the same elements. Consequently, they must have the same cardinality, regardless of the order in which their elements are written. Since A = B, we can write n(A) = n(B). The given value n(A) = 5 therefore immediately gives n(B) = 5. No additional information is needed.

Set order does not matter, so B contains the elements 1, 2, and 3. For A to equal B, it must contain exactly these same elements. Since 1 and 2 are already present in A, the remaining element represented by a must be 3. Values 1 or 2 would create repetition rather than supply the missing element, and 0 would make the sets unequal.

The prime natural numbers less than 10 are 2, 3, 5, and 7. Each of these numbers has exactly two positive factors, while 1 is neither prime nor composite and 4, 6, 8, and 9 are composite. Thus the descriptive set has precisely the elements listed in A, so the two sets are equal.

For an integer x, the inequality |x| < 3 means that x lies strictly between −3 and 3. The integers in this interval are −2, −1, 0, 1, and 2. The endpoints −3 and 3 are excluded because the inequality is strict. Thus A = {−2, −1, 0, 1, 2}, which contains five distinct elements.

A positive divisor of a number is a positive integer that divides it exactly. The only positive integer that divides 1 is 1 itself, because 1 ÷ 1 = 1. Thus the set of positive divisors of 1 is {1}. A set containing exactly one element is called a singleton, and because its elements can be counted, it is also finite.

The set ℕ denotes the natural numbers, which are non-negative in the convention used here, or positive numbers in conventions that begin with 1. In either convention, no negative number belongs to ℕ. Therefore there is no element satisfying the condition “x is a negative natural number.” The required set has no elements and is consequently the empty set, written ∅.

Using the standard school convention, the vowels in the English alphabet are a, e, i, o, and u. Sometimes y is treated as a vowel in particular words, but it is not included in the usual basic alphabet classification. In either interpretation, the number of vowels is limited. Therefore, the set of vowels is finite, not empty or infinite.

By definition, a triangle is a polygon with exactly three sides. A figure with four sides is a quadrilateral, not a triangle. Consequently, no geometric object can satisfy both conditions “is a triangle” and “has four sides.” Since the defining condition produces no elements, the set of such triangles is empty, denoted by ∅.

Two sets are equal when they contain exactly the same elements, regardless of the order in which those elements are written. A and B both contain a, b, and c, so A = B. Set C contains only a and b; it does not contain c. Therefore, C is not equal to A, and the correct statement is A = B and A ≠ C. None of these sets is empty.

The positive natural-number factors of 18 are 1, 2, 3, 6, 9, and 18. Thus A = {1, 2, 3, 6, 9, 18}. This list is complete because every factor of 18 occurs in one of the factor pairs 1×18, 2×9, or 3×6. Since A has six countable elements, it is a finite set.

The natural numbers not exceeding 12 that are divisible by 3 are 3, 6, 9, and 12. Therefore B = {3, 6, 9, 12}, which is exactly the set A. Two sets are equal when they contain precisely the same elements; the order or the way in which the elements are described does not affect equality. Hence A = B.

The condition requires a natural number strictly greater than 5 and strictly less than 6. Since 5 and 6 are consecutive natural numbers, there is no natural number satisfying both inequalities. Thus A contains no element and is the empty set, written as ∅. It is not a singleton because no value meets the stated condition.

Since x must be an integer and must lie between −3 and 3, the elements of C are −3, −2, −1, 0, 1, 2, and 3. Counting them gives seven distinct elements. Therefore, C is a finite set and its cardinality, denoted by n(C), is 7. The inclusive signs ≤ ensure that both boundary values, −3 and 3, are included.

The odd natural numbers begin 1, 3, 5, 7, 9, and continue indefinitely. For every odd natural number, adding 2 produces another odd natural number, so there can never be a final element. The set therefore contains infinitely many elements. Option D lists only some initial elements, not the complete set.

A set is determined by its elements, not by the order in which those elements are written. Both C and D contain exactly the four distinct elements 2, 4, 6, and 8. Reversing their display order does not create a new set. Therefore C and D are equal. The other options incorrectly treat order as important or misclassify a clearly finite set.

The positive multiples of 5 are 5, 10, 15, 20, 25, and so on. For every listed positive multiple, adding 5 gives another positive multiple, so the sequence has no final term. Because the set continues without end, it contains infinitely many elements and is an infinite set. It is neither empty nor finite.

The months in a year are January through December, giving exactly 12 distinct elements. Since this list has a fixed number of members and the counting ends after December, the set is finite. The set is not empty, not infinite, and “equal set” describes a relationship between two sets rather than the size of one set.

Repeated writing of an element does not increase the number of distinct elements in a set. In H, the symbol a appears twice, but it is still only one distinct element. Thus H can be rewritten as {a, b, c}, which is exactly G. Therefore G and H are equal sets, and H does not have four distinct elements.

The set contains integers strictly greater than 2 and strictly less than 3. The only numbers between 2 and 3 are non-integers, while there is no integer in this open interval. Therefore, no integer satisfies the condition, so the set has no elements and is the empty set, written as ∅.

By definition, an empty set is a set containing no element. Therefore, its cardinality, or number of elements, is zero, written as n(∅) = 0. It is not infinite. Also, the order in which elements are written does not matter in a set, so equal sets need not have the same arrangement.