Power Set and Subsets

A set with n elements has 2ⁿ subsets because each element can either be included or excluded. For three elements, the total number of subsets is 2³ = 8. A proper subset is any subset other than the set itself, so we exclude exactly one subset, namely the original set. Therefore, the number of proper subsets is 8 − 1 = 7.

R denotes the set of real numbers, while Z denotes the set of integers. The number 1/2 is real, so every element of {1/2} belongs to R. However, 1/2 is not an integer, so this set is not a subset of Z. The other listed sets contain only integers and are subsets of both R and Z.

A singleton is a set containing exactly one element. Since √2 is a real number, the set {√2} is a subset of R and has precisely one member. The empty set has zero elements, the interval (0, 1) has infinitely many real elements, and {1, 2} has two elements. Thus option A satisfies both conditions.

The symbol ℤ denotes the set of all integers: ..., -2, -1, 0, 1, 2, 3, .... A set is a subset of ℤ only when every one of its elements is an integer. The elements -2, 0, and 5 are all integers, so {-2, 0, 5} is a subset of ℤ. The other choices contain a fraction, an irrational number, or a non-integer decimal.

A power set contains every subset of the original set, including the empty set and the set itself. For a set with n elements, the number of subsets is 2^n because each element can either be included or excluded. Here A has 2 elements, so |P(A)| = 2^2 = 4. The subsets are ∅, {1}, {2}, and {1, 2}.

A proper subset contains only elements of the original set and is not equal to the original set. The set {2, 4} contains elements that are both in A, and it has fewer elements than A, so it is a proper subset. Option A is equal to A, while options C and D contain elements, 10 or 1, that are not in A.

A one-element subset, also called a singleton set, contains exactly one element from the original set. Selecting 1, 2, or 3 individually from A gives the three singleton subsets {1}, {2}, and {3}. The empty set has zero elements, the two-element choices in option B are not singletons, and A itself has three elements, so option A is complete and correct.

A zero-element subset contains no elements, so it must be the empty set ∅. There is exactly one empty set, and the empty set is a subset of every set, including A = {1, 2, 3, 4}. Therefore A has exactly one subset with zero elements. This also agrees with the formula for choosing zero elements: C(4,0) = 1.

For a finite set with n distinct elements, each element has two independent choices in forming a subset: it is either selected or not selected. Therefore the total number of subsets is 2ⁿ. Here A has three elements, so the number of subsets is 2³ = 8. These include the empty set, the three one-element subsets, the three two-element subsets, and A itself.

R denotes the set of real numbers, and every integer is a real number. Consequently, every element of Z belongs to R, so Z ⊆ R. The number i is the imaginary unit, and expressions such as i and 2 + i are non-real complex numbers. Thus the imaginary-number choices are not subsets of R, making A the only correct answer.

For a set containing n distinct elements, each element has exactly two independent choices when forming a subset: it is either included or excluded. Therefore, the total number of subsets is 2ⁿ. With n = 4, the number is 2⁴ = 2 × 2 × 2 × 2 = 16. This count includes both the empty set and the original set itself, so the correct answer is 16.

A set containing n elements has 2^n subsets because each element can either be included or excluded. For n = 2, the total number of subsets is 2^2 = 4. These are the empty set, the two one-element subsets, and the original set. A proper subset is any subset other than the set itself, so the number of proper subsets is 4 - 1 = 3. Therefore, option C is correct.

A set B is a subset of A when every element of B is also an element of A. In option A, both 5 and 9 belong to A = {5, 7, 9}, so {5, 9} is a subset of A. Option B contains 8, option C contains 10, and option D contains 4; none of these numbers belongs to A. Thus option A is the only correct answer.

The power set P(A) is the set of all subsets of A. For A = {1, 2}, its power set is P(A) = {∅, {1}, {2}, {1, 2}}. Therefore {1} is an element of P(A), because it is a subset of A. The number 3, the set {3}, and the set {1, 3} are not included because 3 is not an element of A.

If a finite set has n elements, then each element has two choices in forming a subset: it is either included or excluded. Hence the total number of subsets is 2ⁿ. Here 2ⁿ = 16, and 16 = 2⁴. Therefore n = 4, so the set has four elements. This count includes both the empty set and the original set, as both are always subsets of a set.

Set A has four elements, so its total number of subsets is 2⁴ = 16. A proper subset is any subset that is not equal to the original set A. The original set itself is one of the 16 subsets, so it must be excluded when counting proper subsets. Thus the number of proper subsets is 16 − 1 = 15. The empty set is included among these proper subsets.

A set X is a subset of A when every element of X is also an element of A. The empty set, {1}, and {2} all satisfy this condition. However, {1, 3} contains 3, and 3 is not an element of A = {1, 2}. Therefore {1, 3} is not a subset, making option D correct.

Using the usual school convention N = {1, 2, 3, ...}, the condition x < 5 gives A = {1, 2, 3, 4}. Every element of A occurs in B, so A is a subset of B. However, B also contains 5, which is not in A; therefore the two sets are not equal and B is not a subset of A. Hence option C is correct.

The closed interval A = [0, 10] contains every real number from 0 through 10, including both endpoints. For a set to be a subset of A, each one of its elements must lie in that interval. The numbers 0, 5, and 10 all belong to A, so {0, 5, 10} is a subset. The other choices contain −1 or numbers greater than 10, so they cannot be subsets.

An interval is a subset of [2, 8] only when every one of its points lies between 2 and 8, including endpoint conditions. The interval [3, 7] is entirely contained inside [2, 8], and both of its endpoints are valid members of the larger interval. Option A includes numbers below 2, option C includes numbers below 2, and option D includes numbers greater than 8. Therefore option B is correct.