Disconnected Sets Problem-Solving Quiz

1) If a set in ℝ contains two nonempty disjoint open subsets whose union is the set, it is disconnected

True

False

This is the definition of a disconnected set.

Explanation

This is the definition of a disconnected set.

2) Every finite subset of ℝ with more than one point is connected.

True

False

Finite sets have isolated points → disconnected.

Explanation

Finite sets have isolated points → disconnected.

3) Removing a single point from a connected set always produces a disconnected set

True

False

Removing an endpoint keeps (0,1] connected.

Explanation

Removing an endpoint keeps (0,1] connected.

4) A set consisting of a countably infinite number of isolated points is disconnected.

True

False

Isolated points create many separate pieces.

Explanation

Isolated points create many separate pieces.

5) A disconnected set can have exactly one component.

True

False

Disconnected sets must have at least two components.

Explanation

Disconnected sets must have at least two components.

6) Which of the following sets in ℝ is disconnected?

[0,1] ∪ [2,3]

[0,3]

(1,2)

[0,2]

There is a gap between [0,1] and [2,3].

Explanation

There is a gap between [0,1] and [2,3].

7) Let A = {x ∈ ℝ : x ≤ -1 or x ≥ 1}. Which statement is true?

A is connected

A is disconnected

A is empty

A is compact

A = (−∞,−1] ∪ [1,∞) → two separated parts.

Explanation

A = (−∞,−1] ∪ [1,∞) → two separated parts.

8) Which of these sets is connected?

{0,1,2}

[0,1]

(1,2) ∪ (3,4)

{1/n}

[0,1] is a single interval; all others have gaps.

Explanation

[0,1] is a single interval; all others have gaps.

9) Let X = {x ∈ ℝ : x² < 1 or x² > 4}. Which is true?

X is connected

X is disconnected

X is empty

X is a single interval

X = (−1,1) ∪ (−∞,−2) ∪ (2,∞).

Explanation

X = (−1,1) ∪ (−∞,−2) ∪ (2,∞).

10) A set S ⊂ ℝ² consisting of two disjoint circles is:

Connected

Disconnected

Empty

Compact but connected

Two separate circles → disconnected.

Explanation

Two separate circles → disconnected.

11) Identify all disconnected sets:

{1,2,3}

[0,1]

(1,2) ∪ (3,4)

[0,2]

A is finite with isolated points; C has a gap.

Explanation

A is finite with isolated points; C has a gap.

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12) Which of the following are disconnected?

ℤ

[0,5]

(−∞,−1] ∪ [1,∞)

(0,1)

ℤ is disconnected; (−∞,−1] and [1,∞) have a gap.

Explanation

ℤ is disconnected; (−∞,−1] and [1,∞) have a gap.

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13) Select all disconnected subsets of ℝ²:

Two horizontal segments

Unit disk

Two disjoint circles

Line segment

Separated segments and circles are disconnected.

Explanation

Separated segments and circles are disconnected.

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14) Which sets are disconnected in ℝ?

[0,1] ∪ [2,3]

(−∞,−2] ∪ [1,∞)

{0} ∪ {1/n}

[0,2]

A and B have gaps; C is connected because 1/n → 0.

Explanation

A and B have gaps; C is connected because 1/n → 0.

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15) Which statements are true about disconnected sets?

Removing a point may or may not disconnect

Finite sets >1 point are disconnected

Union of connected sets always disconnected

Disconnected sets always have isolated points

C is false; D is false.

Explanation

C is false; D is false.

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Disconnected Sets Problem-Solving Quiz

1) If a set in ℝ contains two nonempty disjoint open subsets whose union is the set, it is disconnected

2)

What first name or nickname would you like us to use?

2) Every finite subset of ℝ with more than one point is connected.

3) Removing a single point from a connected set always produces a disconnected set

4) A set consisting of a countably infinite number of isolated points is disconnected.

5) A disconnected set can have exactly one component.

6) Which of the following sets in ℝ is disconnected?

7) Let A = {x ∈ ℝ : x ≤ -1 or x ≥ 1}. Which statement is true?

8) Which of these sets is connected?

9) Let X = {x ∈ ℝ : x² < 1 or x² > 4}. Which is true?

10) A set S ⊂ ℝ² consisting of two disjoint circles is:

11) Identify all disconnected sets:

12) Which of the following are disconnected?

13) Select all disconnected subsets of ℝ²:

14) Which sets are disconnected in ℝ?

15) Which statements are true about disconnected sets?