Connected Sets Problem-Solving Quiz

1) Let A=[0,1] and B=[1,2] in ℝ. Which of the following is true?

A∪B is disconnected

A∩B = ∅

A∪B is connected

A and B are disconnected individually

They meet at 1, so [0,2] is a single interval → connected.

Explanation

They meet at 1, so [0,2] is a single interval → connected.

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

(0,1)(2,3)

(−∞,0] ∪ [1,∞)

[−2,0] ∪ {1}

[0,3]

[0,3] is an interval. All others have gaps.

Explanation

[0,3] is an interval. All others have gaps.

3) If S=[0,1] ∪ [1.5,2], which modification will make it connected?

Remove [0,1]

Add [1,1.5]

Add a singleton {0}

Remove [1.5,2]

Adding [1,1.5] fills the gap and creates the full interval [0,2].

Explanation

Adding [1,1.5] fills the gap and creates the full interval [0,2].

4) Let A=(−1,0) ∪ (0,1). What is the connected component of 0?

{0}

(−1,0)

(0,1)

A

0 is not included in either interval, so it stands alone.

Explanation

0 is not included in either interval, so it stands alone.

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

[−1,1) ∪ [2,3]

(−∞,0)

{−1,0,1}

(1,2) ∪ (3,4)

(−∞,0) is an interval. All others have gaps.

Explanation

(−∞,0) is an interval. All others have gaps.

6) The union of connected sets A=[0,1], B=[2,3], C=[1,2] is:

Disconnected

Connected

Connected only if A and B are open

Cannot be determined

All three join to form [0,3], a single interval.

Explanation

All three join to form [0,3], a single interval.

7) Which of the following intervals is not connected in ℝ?

(0,5)

[−3,1]

(1,2)(2,3)

[0,1]

(1,2) ∪ (2,3) has a gap at 2.

Explanation

(1,2) ∪ (2,3) has a gap at 2.

8) Which of the following sets is connected?

[0,1) ∪ {2}

(−∞,−1] ∪ [0,∞)

(−2,2)

{1,2,3}

(−2,2) is an interval; the others split into separate pieces.

Explanation

(−2,2) is an interval; the others split into separate pieces.

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

(0,1)

[0,1]

[0,1) ∪ (1,2]

{0}

(0,1) and [0,1] are intervals; {0} is a singleton.

Explanation

(0,1) and [0,1] are intervals; {0} is a singleton.

Submit

10) Identify the connected sets:

[−3,0]

[1,2] ∪ [3,4]

(−∞,0)

{−1,0,1}

Both are intervals; the others have gaps.

Explanation

Both are intervals; the others have gaps.

Submit

11) For the sets A=[0,2], B=[2,4], C=[5,6], which unions are connected?

AB

BC

AC

ABC

A and B meet at 2 → connected.

Explanation

A and B meet at 2 → connected.

12) Which of the following sets are connected?

(−1,0) ∪ [0,1]

[1,2) ∪ (2,3]

(−∞,−1]

{2}

A is the full interval (−1,1]; C is an interval; D is a singleton.

Explanation

A is the full interval (−1,1]; C is an interval; D is a singleton.

Submit

13) Which sets are disconnected?

[0,1] ∪ [2,3]

(−2,0]

{0,1}

(0,5)

Both have gaps; the others are intervals.

Explanation

Both have gaps; the others are intervals.

Submit

14) If A=(−1,1), B=[1,2], C=[2,3], which combinations form a connected union?

AB

BC

AC

ABC

A touches B at 1; B touches C at 2; ABC forms (−1,3].

Explanation

A touches B at 1; B touches C at 2; ABC forms (−1,3].

Submit

15) Identify connected sets in ℝ:

[0,1] ∪ (1,2]

(−∞,0]

{1,2,3}

(0,5)

Both are intervals; A and C contain gaps.

Explanation

Both are intervals; A and C contain gaps.

Submit

Connected Sets Problem-Solving Quiz

1) Let A=[0,1] and B=[1,2] in ℝ. Which of the following is true?

2)

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

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

3) If S=[0,1] ∪ [1.5,2], which modification will make it connected?

4) Let A=(−1,0) ∪ (0,1). What is the connected component of 0?

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

6) The union of connected sets A=[0,1], B=[2,3], C=[1,2] is:

7) Which of the following intervals is not connected in ℝ?

8) Which of the following sets is connected?

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

10) Identify the connected sets:

11) For the sets A=[0,2], B=[2,4], C=[5,6], which unions are connected?

12) Which of the following sets are connected?

13) Which sets are disconnected?

14) If A=(−1,1), B=[1,2], C=[2,3], which combinations form a connected union?

15) Identify connected sets in ℝ: