Sets and their representations

An open interval (2, 5) contains every real number strictly greater than 2 and strictly less than 5. The round brackets exclude both endpoints. Thus 3, 4, and 4.9 lie inside the interval, but 2 is not included. Therefore, option A is correct. Notice that 5 would also be excluded, but it is not among the listed choices.

In interval notation, a square bracket includes the endpoint and a parenthesis excludes it. The left bracket in [2, 5) includes 2, so x may equal 2. The right parenthesis excludes 5, so x must be less than 5. Therefore the equivalent inequality is 2 ≤ x < 5, making option A correct.

In interval notation, a round bracket excludes an endpoint and a square bracket includes it. The interval (2, 5] therefore excludes 2, giving x > 2, and includes 5, giving x ≤ 5. Combining both conditions produces 2 < x ≤ 5, which is option A. Options B, C, and D either include 2, exclude 5, or do both incorrectly.

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.