Sets and their representations

The inequality 1 < x < 4 requires x to be greater than 1 and less than 4. Neither endpoint is allowed because there is no equality sign at either end. In interval notation, excluded endpoints are written with parentheses. Therefore, the correct interval is (1, 4), which is option A.

The symbol ≥ means greater than or equal to. Thus, the solution contains 3 as well as every real number greater than 3. Since 3 is included, a square bracket is used at the left endpoint. The interval extends toward positive infinity, where a parenthesis is required because infinity is not a real number that can be included. Therefore, the answer is [3, ∞).

The inequality x ≤ 5 includes 5 and every real number less than 5. Because equality is permitted, the endpoint 5 belongs to the solution set and is written with a square bracket. The interval continues indefinitely toward negative infinity, and infinity always uses a parenthesis. Therefore, the correct interval notation is (-∞, 5].

In interval notation, a parenthesis means that the endpoint is excluded, while a square bracket means that the endpoint is included. The interval (0, 4] therefore excludes 0 because of the left parenthesis and includes 4 because of the right square bracket. It contains all real numbers greater than 0 and less than or equal to 4.

An interval includes an endpoint when a square bracket is placed next to that endpoint. In the interval [a, b], square brackets appear on both sides, so both a and b are included. This is called a closed interval. By contrast, (a, b) excludes both endpoints, while [a, b) and (a, b] include only one endpoint each.

Parentheses indicate that the adjacent endpoint is excluded. Therefore, the interval (a, b) contains all real numbers strictly between a and b, but it contains neither a nor b. It is called an open interval. The other options use at least one square bracket, so each of them includes one or both endpoints and cannot be correct.

If a set has n elements, its total number of subsets is 2ⁿ, because each element can either be included in a subset or excluded from it. Here A = {9} has n = 1 element, so the number of subsets is 2¹ = 2. These two subsets are the empty set ∅ and the set {9} itself. Therefore option B is correct.

The interval (-∞, ∞) represents every real number because there is no finite lower or upper boundary. It covers the entire real number line, including negative numbers, zero, positive numbers, rational numbers, and irrational numbers. Therefore it is the set of real numbers, denoted by R. It is not limited to natural numbers, integers, or rational numbers, so option A is correct.

Non-negative real numbers are all real numbers greater than or equal to zero, so the set is {x ∈ R : x ≥ 0}. The number 0 must be included, which requires a square bracket at the left endpoint. There is no largest real number, so infinity is written with a parenthesis. Thus the correct interval is [0, ∞), option A.

Positive real numbers are all real numbers strictly greater than zero. Therefore, zero must be excluded, which is shown by a round parenthesis at 0. There is no greatest positive real number, so the interval extends indefinitely toward positive infinity. Hence the correct notation is (0, ∞), not [0, ∞), because the latter includes zero.

Negative real numbers are precisely the real numbers less than zero. Zero itself is neither negative nor positive, so it must be excluded; this is indicated by a round parenthesis at 0. The values continue without bound toward the left, represented by negative infinity. Therefore, the correct interval is (-∞, 0).

A non-positive real number is less than or equal to zero. Thus the set contains every negative real number as well as zero. The interval extends indefinitely to the left, so it begins at negative infinity, and the square bracket at 0 shows that zero is included. Hence the correct notation is (-∞, 0].

The notation [2, 6] represents a closed interval. It contains every real number between 2 and 6, and the square brackets show that both endpoints are included. Therefore, x may be equal to 2, equal to 6, or any real number between them. The correct inequality is 2 ≤ x ≤ 6, so option A is correct.

The interval (2, 6) is open at both ends because it uses round parentheses. Consequently, 2 and 6 are excluded, while every real number strictly between them is included. The equivalent inequality is therefore 2 < x < 6. Option B would describe the closed interval [2, 6], so it is not correct.

The empty set, written as ∅, has no elements. A set S is a subset of T if every element of S is also in T. Since ∅ has no elements, there is no element that can violate this condition, so ∅ ⊆ T for every set T. The empty set is different from {0}, because {0} contains the single element 0.

The interval (2, 5) is an open interval because round parentheses are used at both ends. Therefore, 2 and 5 are not included, while every real number strictly between them is included. In set-builder notation, the interval is {x : 2 < x < 5}. For example, 3 and 4.9 belong to the interval, but 2 and 5 do not.

The interval [1, 4] is a closed interval because square brackets are used at both ends. A square bracket means that the endpoint is included. Hence, both 1 and 4 belong to this interval, while numbers below 1 or above 4 do not. Among the given choices, 1 is definitely included.

The square bracket in [3, 7) means that the left endpoint 3 belongs to the interval. The round bracket after 7 means that 7 is excluded. In inequality form, the interval is 3 ≤ x < 7, so it contains 3 but not 7. Consequently, option A is correct. The other choices reverse or ignore the meanings of the brackets.

The symbol 0 ≤ x allows x to equal 0, so the lower endpoint is included and must use a square bracket. The symbol x < 6 excludes 6, so the upper endpoint must use a round bracket. Therefore the interval is [0, 6), making option B correct. The other forms incorrectly include or exclude one or both endpoints.

The inequality 2 < x excludes 2, so the interval must begin with a round parenthesis. The condition x ≤ 9 includes 9, so the interval must end with a square bracket. Combining these endpoint rules gives (2, 9], which is option A. Option B includes 2 and excludes 9; C includes both endpoints, while D excludes both.

The inequality x ≥ 5 includes 5 because equality is allowed, so a square bracket is used at 5. It also includes every real number greater than 5, continuing without upper bound toward infinity. Infinity is not an actual real endpoint, so a round bracket is always used with it. Thus the correct interval is [5, ∞).

The power set of a set is the set of all its subsets. Therefore, an element of P({1, 2, 3}) must itself be a set whose every element comes from {1, 2, 3}. The set {1, 3} satisfies this condition, so it belongs to the power set. The single number 1 is not written as a subset here, and 4 is not in the original set.

The condition requires x to be an integer between -2 and 2, with both endpoints included because the inequalities use ≤. Listing the integers in order gives -2, -1, 0, 1, and 2. Thus A = {-2, -1, 0, 1, 2}. Fractions and decimals are not considered, and no endpoint may be omitted.

An interval includes an endpoint when a square bracket is used and excludes it when a round bracket is used. In (1,4], the round bracket at 1 means that 1 is excluded, while the square bracket at 4 means that 4 is included. Option A excludes both endpoints, option B includes 1 but excludes 4, and option D includes both endpoints.

An open interval excludes both of its finite endpoints. Round brackets indicate exclusion, so (2,6) represents all real numbers x satisfying 2<x<6; neither 2 nor 6 belongs to the interval. Option A is closed because both endpoints are included, while options C and D are half-open because exactly one endpoint is included.