Boundedness: Concept Mastery Quiz

1) If a set A in a metric space (X,d) is bounded, then there exists M > 0 such that d(x,y) ≤ M for all x,y ∈ A.

True

False

True, because boundedness means all pairwise distances in A are bounded by some constant.

Explanation

True, because boundedness means all pairwise distances in A are bounded by some constant.

2) Which of the following sets in ℝ with the usual metric is bounded?

[0,5]

(-∞, 3)

ℤ

(0, ∞)

[0,5] lies within a finite interval; the others extend infinitely.

Explanation

[0,5] lies within a finite interval; the others extend infinitely.

3) Every finite subset of a metric space is bounded.

True

False

True, because finitely many distances achieve a maximum value.

Explanation

True, because finitely many distances achieve a maximum value.

4) Let A = {(x,y) ∈ ℝ² : x² + y² ≤ 4}. Which statement is true?

A is bounded but not closed

A is unbounded

A is bounded and closed

A is open and unbounded

It is a closed disk of radius 2, so it is bounded and closed.

Explanation

It is a closed disk of radius 2, so it is bounded and closed.

5) If a set A in a metric space is bounded, then every subset of A is also bounded.

True

False

True, because all distances in a subset are distances from A, which are already bounded.

Explanation

True, because all distances in a subset are distances from A, which are already bounded.

6) Let (X,d) be a metric space and let A, B ⊂ X be bounded. Which must be true?

A ∩ B is bounded

A ∪ B is bounded

Both A and B

None of the above

The intersection is contained in each set, so it inherits boundedness. The union may not be bounded.

Explanation

The intersection is contained in each set, so it inherits boundedness. The union may not be bounded.

7) A set A is bounded in a metric space (X,d) if there exists x₀ ∈ X and r > 0 such that A ⊂ B(x₀,r).

True

False

True, because this is an equivalent definition of boundedness.

Explanation

True, because this is an equivalent definition of boundedness.

8) Which set is NOT necessarily bounded?

The union of finitely many bounded sets

The intersection of bounded sets

The Cartesian product of two bounded sets in ℝ

The set of all rational numbers between 0 and 1

A bounded × bounded set in ℝ may not be bounded under some metrics.

Explanation

A bounded × bounded set in ℝ may not be bounded under some metrics.

9) If a sequence (xₙ) in a metric space is bounded, then the set {xₙ : n ∈ ℕ} is bounded.

True

False

True, because boundedness of the sequence means all its elements lie in some bounded ball.

Explanation

True, because boundedness of the sequence means all its elements lie in some bounded ball.

10) Let A = { xₙ = n/(n+1) : n ∈ ℕ } in ℝ. Which is true?

A is bounded above but not below

A is bounded below but not above

A is bounded both above and below

A is unbounded

All values lie between 0 and 1.

Explanation

All values lie between 0 and 1.

11) A bounded set in a metric space is always compact.

True

False

False, compactness also requires closedness and completeness.

Explanation

False, compactness also requires closedness and completeness.

12) Which condition implies A ⊂ X is bounded?

A is finite

A is contained in some open ball

A is a subset of a compact set

All of the above

Any of these conditions ensures boundedness.

Explanation

Any of these conditions ensures boundedness.

13) If a set A ⊂ ℝ is bounded and its supremum exists, then sup A ∈ A.

True

False

False, sup A need not belong to A.

Explanation

False, sup A need not belong to A.

14) Let Aₙ = { x ∈ ℝ : |x| < n } for n∈N . Which statement is true?

Each Aₙ is unbounded

⋂ Aₙ is bounded

Each Aₙ is bounded, but ⋃ Aₙ is unbounded

⋂ Aₙ is unbounded

Each Aₙ is bounded, but their union is ℝ, unbounded.

Explanation

Each Aₙ is bounded, but their union is ℝ, unbounded.

15) The empty set is considered bounded in any metric space.

True

False

True, because boundedness requires ∃M such that all distances in A are ≤ M, which is vacuously true for the empty set.

Explanation

True, because boundedness requires ∃M such that all distances in A are ≤ M, which is vacuously true for the empty set.