Q: The set {x : x ∈ ℝ, −1 ≤ x ≤ 4} is equal to which interval?

[−1, 4] — both endpoints included The symbol ≤ includes equality, so both boundary values are part of the set. The condition −1 ≤ x includes −1, and x ≤ 4 includes 4. In interval notation, an included endpoint is written with a square bracket. Since both endpoints are included, the set is the closed interval [−1, 4].

Q: If n(U) = 60 and n(A ∪ B) = 45, how many elements are in (A ∪ B)'?

15 The complement of A ∪ B consists of all elements in the universal set U that are not in the union. For a finite universal set, n((A ∪ B)') = n(U) − n(A ∪ B). Substituting the values gives 60 − 45 = 15. Thus, option A is correct. In a Venn diagram, this is the area of the rectangle outside both circles.

Question 1

Which statement is true for A = {0}?

Accepted Answer

It is a singleton and finite set The braces in A = {0} show that 0 is an element of the set. There is exactly one distinct element, so A is a singleton set. A set containing one element is also finite. It is not empty, because the empty set contains no elements, and {0} is not equal to ∅. Therefore, option B is correct.

Question 2

If A = {x ∈ ℕ : x² − 5x + 6 = 0}, which set is equal to A?

Accepted Answer

{2, 3} Find the natural-number solutions by factoring the quadratic: x² − 5x + 6 = (x − 2)(x − 3). Hence x = 2 or x = 3. Both values are natural numbers, so the set defined by the condition is A = {2, 3}. Therefore, option B is equal to A. Option A uses incorrect roots, option C gives unrelated negative values, and option D wrongly treats the solution set as empty.

Question 3

The interval [1, 1] is equal to which set?

Accepted Answer

{1} The closed interval [1, 1] contains all real numbers x satisfying 1 ≤ x ≤ 1. The only real number that satisfies both inequalities is x = 1, and the square brackets indicate that the endpoint is included. Therefore [1, 1] contains exactly one element and is equal to the singleton set {1}. Option A is correct.

Question 4

The interval (1, 1) is equal to what?

Accepted Answer

∅ The open interval (1, 1) consists of real numbers x satisfying 1 < x < 1. No real number can be simultaneously greater than 1 and less than 1. The parentheses also exclude both endpoints, so 1 is not included. Consequently the interval contains no elements and is the empty set ∅. Hence option A is correct.

Question 5

The set {x : x ∈ ℝ, −1 ≤ x ≤ 4} is equal to which interval?

Accepted Answer

[−1, 4] — both endpoints included The symbol ≤ includes equality, so both boundary values are part of the set. The condition −1 ≤ x includes −1, and x ≤ 4 includes 4. In interval notation, an included endpoint is written with a square bracket. Since both endpoints are included, the set is the closed interval [−1, 4].

Question 6

If n(U) = 60 and n(A ∪ B) = 45, how many elements are in (A ∪ B)'?

Accepted Answer

15 The complement of A ∪ B consists of all elements in the universal set U that are not in the union. For a finite universal set, n((A ∪ B)') = n(U) − n(A ∪ B). Substituting the values gives 60 − 45 = 15. Thus, option A is correct. In a Venn diagram, this is the area of the rectangle outside both circles.

The Empty Set, many more. Step 3: Hence the set is infinite.

Practice questions

01 Which statement is true for A = {0}?

02 If A = {x ∈ ℕ : x² − 5x + 6 = 0}, which set is equal to A?

03 The interval [1, 1] is equal to which set?

04 The interval (1, 1) is equal to what?

05 The set {x : x ∈ ℝ, −1 ≤ x ≤ 4} is equal to which interval?

06 If n(U) = 60 and n(A ∪ B) = 45, how many elements are in (A ∪ B)'?

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