Closure Of Sets Properties Quiz

1) The closure of the intersection of two sets equals the intersection of their closures.

True

False

Generally, (A ∩ B)̄ ⊆ Ā ∩ B̄, but not equal.

Explanation

Generally, (A ∩ B)̄ ⊆ Ā ∩ B̄, but not equal.

2) The closure of the complement of a set equals the complement of its interior.

True

False

Always true: closure(Aᶜ) = (interior(A))ᶜ.

Explanation

Always true: closure(Aᶜ) = (interior(A))ᶜ.

3) If set A has a limit point x, then x is in A.

True

False

Limit points need not lie in A (example: A=(0,1), limit point=0).

Explanation

Limit points need not lie in A (example: A=(0,1), limit point=0).

4) The closure of an infinite set always contains infinitely many points.

True

False

An infinite set’s closure cannot collapse to a finite set.

Explanation

An infinite set’s closure cannot collapse to a finite set.

5) In ℝ, the closure of (0,1) ∪ (1,2) is [0,2].

True

False

The endpoints 0, 1, and 2 are all limit points, so closure = [0,2].

Explanation

The endpoints 0, 1, and 2 are all limit points, so closure = [0,2].

6) In ℝ², the closure of the set ({(x, 1/x) : x > 0}) contains the point (0,∞).

True

False

No finite point approaches (0,∞); the graph has no real limit point at x = 0.

Explanation

No finite point approaches (0,∞); the graph has no real limit point at x = 0.

7) Which of the following best defines a closure of a set A?

The largest open set contained in A

The smallest closed set containing A

The set of interior points of A

The complement of A

The closure is the smallest closed set that contains A.

Explanation

The closure is the smallest closed set that contains A.

8) The closure of a set always includes:

Only its interior points

Only its boundary points

The set itself

None of the above

The closure contains A plus its limit points, so it always contains A itself.

Explanation

The closure contains A plus its limit points, so it always contains A itself.

9) The closure of the empty set ∅ is:

ℝ

{0}

∅

Undefined

The empty set has no limit points, so its closure is ∅.

Explanation

The empty set has no limit points, so its closure is ∅.

10) The closure of an open set is:

Must be open

Must be closed

Must be empty

Is always equal to the interior

Closures are always closed sets.

Explanation

Closures are always closed sets.

11) The closure of (0,1) ∪ {5} is:

(0,1)

[0,1]

[0,1] ∪ {5}

[0,1] ∪ (5,∞)

The closure includes 0 and 1 (endpoints) and the isolated point 5.

Explanation

The closure includes 0 and 1 (endpoints) and the isolated point 5.

12) If A ⊆ B, then:

A ⊆ B

A = B

A ⊇ B

No relation exists

If A is a subset of B, then A ⊆ B is the correct statement.

Explanation

If A is a subset of B, then A ⊆ B is the correct statement.

13) Which of the following is always true?

A ∪ B = A ∪ B

A ∪ B = A ∩ B

A ∪ B = A ∪ B

A ∪ B = ∅

A set union equals itself; this is always true.

Explanation

A set union equals itself; this is always true.

14) A set with no limit points has its closure equal to the set itself.

True

False

Such a set contains all its (nonexistent) limit points, so the closure adds nothing.

Explanation

Such a set contains all its (nonexistent) limit points, so the closure adds nothing.

15) The closure of the union of two sets equals the union of their closures.

True

False

Closure distributes over unions: (Ā ∪ B̄).

Explanation

Closure distributes over unions: (Ā ∪ B̄).

Closure of Sets Properties Quiz

1) The closure of the intersection of two sets equals the intersection of their closures.

2)

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

2) The closure of the complement of a set equals the complement of its interior.

3) If set A has a limit point x, then x is in A.

4) The closure of an infinite set always contains infinitely many points.

5) In ℝ, the closure of (0,1) ∪ (1,2) is [0,2].

6) In ℝ², the closure of the set ({(x, 1/x) : x > 0}) contains the point (0,∞).

7) Which of the following best defines a closure of a set A?

8) The closure of a set always includes:

9) The closure of the empty set ∅ is:

10) The closure of an open set is:

11) The closure of (0,1) ∪ {5} is:

12) If A ⊆ B, then:

13) Which of the following is always true?

14) A set with no limit points has its closure equal to the set itself.

15) The closure of the union of two sets equals the union of their closures.