The Empty Set, Finite and Infinite Sets, Equal Sets

A set records membership, not the number of times an element is written. Repeated occurrences of 2 and 3 are counted only once. Thus the given set is equivalent to {1, 2, 3}. It has three distinct elements, so its cardinality, denoted by n(A), is 3. Therefore, option A is correct.

The counting numbers from 1 through 5 are 1, 2, 3, 4, and 5. Thus, A has exactly five distinct elements. A set with a fixed and limited number of elements is called a finite set. It is not empty because it has elements, not infinite because its elements can be completely counted, and not an equal set because equality requires comparison with another set.

An empty set is a set that contains no elements. Its standard symbol is ∅, and it may also be written as { }. The symbol 0 represents the number zero, not a set. The set {0} contains one element, namely zero, so it is not empty. Similarly, 1 is a number rather than the notation for an empty set.

The names of the months in a year are January through December, giving exactly twelve distinct elements. Since the number of month names is fixed and limited, A is a finite set. It is not empty because month names exist, not infinite because there are only twelve of them, and not a singleton because a singleton has exactly one element.

Evaluate the rule defining B for each allowed value of x: 1² = 1, 2² = 4, and 3² = 9. Therefore, B = {1, 4, 9}. Set equality depends on having exactly the same elements, regardless of how the sets are represented or ordered. Since A has precisely these elements, A = B.

The empty set ∅ has no elements. There is no natural number less than zero, so the condition in option C cannot be satisfied by any natural number. Consequently, that set contains no elements and is equal to ∅. In contrast, {0}, {1}, and {∅} each contain one element; notably, {∅} contains the empty set as an element and is not itself empty.

The positive multiples of 3 that are less than 15 are 3, 6, 9, and 12. The number 15 is excluded because the condition says less than 15, not less than or equal to 15. Therefore, A = {3, 6, 9, 12}, which has four elements. The set is finite because this list ends after a limited number of values.

Two sets are equal precisely when they contain the same elements. The order in which elements are written does not matter, and repeated listing of an element does not create a new element. Equal sets need not be empty; for example, {1, 2} and {2, 1} are equal. Hence, option B is correct.

A prime number has exactly two positive divisors: 1 and itself. Among all even numbers, only 2 has this property; every other even number is divisible by 2 and another integer, so it is not prime. Hence the set of even prime numbers is A = {2}. It contains one element, so it is also a singleton and a finite set.

The natural numbers greater than 50 are 51, 52, 53, 54, and so on. There is no greatest natural number, so new members can always be found beyond any listed value. The set therefore continues without end and has infinitely many elements. Hence, the correct answer is option D, infinite.

Sets are equal when they have exactly the same distinct elements; the order does not matter, and repeated elements are counted only once. Thus options A and D contain the same elements as A, while option B also represents {1, 2, 3} because 1 is repeated. Option C has the additional element 4, so it is not equal.

Solving x + 1 = 1 gives x = 0. Under the usual school convention used here, natural numbers are 1, 2, 3, ...; therefore 0 is not a natural number. No value satisfies both conditions simultaneously, so A contains no element and is the empty set, written as ∅. Thus option C is correct.

The empty set, denoted by ∅, contains no elements, so its cardinality is 0. A set with a definite, limited number of elements is finite; therefore a set with zero elements is also finite. The sets in options C and D continue without end and are infinite, while {0} is finite but not empty. Hence option A is correct.

A positive divisor of 6 is a positive integer that divides 6 exactly, with zero remainder. The factor pairs 1 × 6 and 2 × 3 show that the positive divisors are 1, 2, 3, and 6. No other positive integer divides 6 without a remainder. Thus the roster form is A = {1, 2, 3, 6}, so option A is correct. Options B and C omit valid divisors, while D incorrectly treats the set as empty.

An infinite set has endlessly many distinct elements, whereas a finite set has a fixed number of elements. The odd natural numbers and the integers continue without end, and the positive multiples of 5 are also unlimited. Natural numbers less than 10 are only {1, 2, 3, 4, 5, 6, 7, 8, 9}, so this is a finite set. Therefore, option C is correct.

The positive even numbers less than 8 are 2, 4, and 6. Thus the condition defining B produces B = {2, 4, 6}, which contains exactly the same elements as A. Two sets are equal when they have precisely the same elements, regardless of how they are written. Therefore A = B and option A is correct.

By definition, a triangle is a polygon with exactly three sides. An object cannot simultaneously be a triangle and have four sides, so no object satisfies the condition defining A. A set with no elements is called the empty set and is represented by ∅. Hence A is empty, and option A is correct.

A perfect-square natural number is a natural number that can be written as the square of a natural number. The natural numbers less than 4 are 1, 2, and 3. Of these, only 1 is a perfect square because 1 = 1². Although 4 = 2², it is not included because the condition is strictly less than 4. Therefore A = {1}, so option A is correct; the other options either include excluded numbers or wrongly make the set empty.

The notation n(A) denotes the cardinality of set A, meaning the number of distinct elements in it. The empty set has 0 elements, {5} has 1 element, {5, 7} has 2 distinct elements, and {1, 2, 3} has 3 elements. Therefore, the set satisfying n(A) = 2 is {5, 7}, which is option C.

In the usual weekly order, Sunday is followed immediately by Monday. There is no named day between these two consecutive days. Consequently, no element satisfies the condition defining A, so A has zero elements and is the empty set. Therefore option A is correct; it is not a singleton or an infinite set.

The odd natural numbers in the inclusive interval from 1 through 10 are numbers that are not divisible by 2: 1, 3, 5, 7, and 9. The number 10 is even, and 11 lies outside the stated interval. Hence the roster form of the set is A = {1, 3, 5, 7, 9}, so option A is correct.

In a set, each element is listed only once; repetition does not create a new element. Therefore, {b, a, a} has the same distinct elements as {a, b}, namely a and b. The order of elements also does not matter in a set. Hence both sets contain exactly the same elements and are equal. Option A is correct.

Assuming natural numbers begin with 1, the members of A are 1, 2, 3, ..., 99. This list contains exactly 99 elements and has a final member, so it cannot continue indefinitely. A set with a fixed, countable number of elements is called a finite set. Therefore, A is finite.

The integers strictly between 14 and 16 contain only 15. However, 15 is not prime because it has divisors 1, 3, 5, and 15; in particular, it is divisible by 3 and 5. Thus no number satisfies both conditions in the set-builder description. Therefore A has no elements and A = ∅.

A set records distinct elements only once, so repeated writing does not increase its membership. The set {0,0,0} therefore contains just one distinct element, namely 0. It is exactly the same set as {0}. The empty set has no elements, and {1} contains 1 rather than 0, so neither of those is equal to {0}.