Identifying Connected Sets Quiz

1) The interval (0,1) ∪ (1,2) is connected in ℝ.

True

False

There is a gap at 1; the set splits into two separate intervals.

Explanation

There is a gap at 1; the set splits into two separate intervals.

2) A half-open interval [0,2) in ℝ is connected.

True

False

Any interval—open, closed, or half-open—is connected.

Explanation

Any interval—open, closed, or half-open—is connected.

3) The union of two connected sets that do not intersect is always connected.

True

False

If they do not touch or overlap, the union has a gap → disconnected.

Explanation

If they do not touch or overlap, the union has a gap → disconnected.

4) Every bounded interval in ℝ is connected.

True

False

All intervals in ℝ are connected, regardless of being bounded or unbounded.

Explanation

All intervals in ℝ are connected, regardless of being bounded or unbounded.

5) The set of all integers ℤ in ℝ is connected.

True

False

ℤ is a discrete set with gaps everywhere → disconnected.

Explanation

ℤ is a discrete set with gaps everywhere → disconnected.

6) Any set consisting of a single point is connected.

True

False

A one-point set cannot be split into two nonempty parts.

Explanation

A one-point set cannot be split into two nonempty parts.

7) Which of the following sets in ℝ is connected?

(−1,0) ∪ (1,2)

[0,1] ∪ {2}

[0,3]

{−1,0,1}

[0,3] is a full interval; all others have gaps.

Explanation

[0,3] is a full interval; all others have gaps.

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

(2,5)

[−2,0]

(0,1) ∪ (2,3)

[1,4]

There is a gap between 1 and 2.

Explanation

There is a gap between 1 and 2.

9) The set [0,1) ∪ [1,2] is:

Connected

Disconnected

Connected only if open

Connected only if closed

The sets meet at 1, forming interval [0,2].

Explanation

The sets meet at 1, forming interval [0,2].

10) Which property must a connected set in ℝ satisfy?

Must be bounded

Must be an interval

Must be open

Must be closed

Connected subsets of ℝ are exactly intervals.

Explanation

Connected subsets of ℝ are exactly intervals.

11) Which of the following sets are connected?

(−∞,0]

[1,2] ∪ [3,4]

[−1,1]

{2,4,6}

(−∞,0] and [−1,1] are intervals; the others have gaps.

Explanation

(−∞,0] and [−1,1] are intervals; the others have gaps.

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12) Which of the following sets in ℝ are disconnected?

(0,1) ∪ (2,3)

[−2,0]

{0,1,2}

(−1,1)

(0,1) ∪ (2,3) and {0,1,2} have gaps.

Explanation

(0,1) ∪ (2,3) and {0,1,2} have gaps.

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13) Let A=[0,1], B=[2,3], C=[1,2]. Which unions are connected?

A ∪ B

A ∪ C

B ∪ C

A ∪ B ∪ C

A∪C and B∪C touch; all three form [0,3].

Explanation

A∪C and B∪C touch; all three form [0,3].

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14) Which of the following sets in ℝ are connected?

(−1,2)

[3,5)

{0}

[0,1] ∪ [2,3]

All are intervals or singletons; [0,1]∪[2,3] has a gap.

Explanation

All are intervals or singletons; [0,1]∪[2,3] has a gap.

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15) Which statements are true regarding connected sets in ℝ?

Any interval is connected

Union of connected sets always connected

Intersection always connected

Discrete sets disconnected

Intervals are connected; discrete multi‑point sets are disconnected.

Explanation

Intervals are connected; discrete multi‑point sets are disconnected.

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Identifying Connected Sets Quiz

1) The interval (0,1) ∪ (1,2) is connected in ℝ.

2)

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

2) A half-open interval [0,2) in ℝ is connected.

3) The union of two connected sets that do not intersect is always connected.

4) Every bounded interval in ℝ is connected.

5) The set of all integers ℤ in ℝ is connected.

6) Any set consisting of a single point is connected.

7) Which of the following sets in ℝ is connected?

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

9) The set [0,1) ∪ [1,2] is:

10) Which property must a connected set in ℝ satisfy?

11) Which of the following sets are connected?

12) Which of the following sets in ℝ are disconnected?

13) Let A=[0,1], B=[2,3], C=[1,2]. Which unions are connected?

14) Which of the following sets in ℝ are connected?

15) Which statements are true regarding connected sets in ℝ?