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Power Sets and Subsets: Unraveling the Structure of Sets

Description: Power Sets and Subsets: Unraveling the Structure of Sets
Number of Questions: 15
Created by:
Tags: set theory power sets subsets cardinality
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Given a set A with n elements, how many subsets does A have?

  1. 2^n

  2. n^2

  3. n!

  4. n

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Correct Option: A
Explanation:

The power set of a set with n elements has 2^n subsets, as each element can be either included or excluded from the subset.

If A and B are two sets, which of the following is always true?

  1. A ∪ B ⊆ P(A ∪ B)

  2. A ∩ B ⊆ P(A ∩ B)

  3. A - B ⊆ P(A - B)

  4. A × B ⊆ P(A × B)

Show answer
Correct Option: A
Explanation:

The union of two sets is always a subset of the power set of their union.

What is the cardinality of the power set of an empty set?

  1. 0

  2. 1

  3. 2

  4. ∞

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Correct Option: B
Explanation:

The power set of an empty set contains only one element: the empty set itself.

If A is a finite set with n elements, how many subsets of A have exactly k elements?

  1. n choose k

  2. n permute k

  3. n combine k

  4. n subtract k

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Correct Option: A
Explanation:

The number of subsets of a set with n elements that have exactly k elements is given by n choose k.

Which of the following is a proper subset of the set {1, 2, 3, 4, 5}?

  1. {1, 2, 3, 4, 5}

  2. {1, 2, 3}

  3. {1, 3, 5}

  4. {2, 4, 5}

Show answer
Correct Option: C
Explanation:

A proper subset is a subset that is not equal to the original set. {1, 3, 5} is a proper subset of {1, 2, 3, 4, 5} because it is a subset and it is not equal to the original set.

If A and B are two sets, which of the following is always true?

  1. P(A ∩ B) = P(A) ∩ P(B)

  2. P(A ∪ B) = P(A) ∪ P(B)

  3. P(A - B) = P(A) - P(B)

  4. P(A × B) = P(A) × P(B)

Show answer
Correct Option: A
Explanation:

The power set of the intersection of two sets is equal to the intersection of the power sets of the two sets.

What is the cardinality of the power set of a set with n elements?

  1. n

  2. 2^n

  3. n!

  4. ∞

Show answer
Correct Option: B
Explanation:

The cardinality of the power set of a set with n elements is 2^n.

If A is a set with n elements, how many subsets of A have an even number of elements?

  1. 2^(n-1)

  2. 2^n

  3. n choose (n/2)

  4. n permute (n/2)

Show answer
Correct Option: A
Explanation:

The number of subsets of a set with n elements that have an even number of elements is 2^(n-1).

Which of the following is a subset of the set {1, 2, 3, 4, 5}?

  1. {1, 2, 3, 4, 5}

  2. {1, 3, 5}

  3. {2, 4}

  4. {6, 7, 8}

Show answer
Correct Option: B
Explanation:

A subset is a set that is contained within another set. {1, 3, 5} is a subset of {1, 2, 3, 4, 5} because every element of {1, 3, 5} is also an element of {1, 2, 3, 4, 5}.

If A and B are two sets, which of the following is always true?

  1. P(A ∪ B) = P(A) ∪ P(B)

  2. P(A ∩ B) = P(A) ∩ P(B)

  3. P(A - B) = P(A) - P(B)

  4. P(A × B) = P(A) × P(B)

Show answer
Correct Option: A
Explanation:

The power set of the union of two sets is equal to the union of the power sets of the two sets.

What is the cardinality of the power set of a set with 3 elements?

  1. 3

  2. 6

  3. 8

  4. 9

Show answer
Correct Option: C
Explanation:

The cardinality of the power set of a set with 3 elements is 8.

If A is a set with n elements, how many subsets of A have an odd number of elements?

  1. 2^(n-1)

  2. 2^n

  3. n choose (n/2)

  4. n permute (n/2)

Show answer
Correct Option: A
Explanation:

The number of subsets of a set with n elements that have an odd number of elements is 2^(n-1).

Which of the following is a proper subset of the set {1, 2, 3, 4, 5}?

  1. {1, 2, 3, 4, 5}

  2. {1, 2, 3}

  3. {1, 3, 5}

  4. {2, 4, 5}

Show answer
Correct Option: C
Explanation:

A proper subset is a subset that is not equal to the original set. {1, 3, 5} is a proper subset of {1, 2, 3, 4, 5} because it is a subset and it is not equal to the original set.

If A and B are two sets, which of the following is always true?

  1. P(A ∩ B) = P(A) ∩ P(B)

  2. P(A ∪ B) = P(A) ∪ P(B)

  3. P(A - B) = P(A) - P(B)

  4. P(A × B) = P(A) × P(B)

Show answer
Correct Option: A
Explanation:

The power set of the intersection of two sets is equal to the intersection of the power sets of the two sets.

What is the cardinality of the power set of a set with n elements?

  1. n

  2. 2^n

  3. n!

  4. ∞

Show answer
Correct Option: B
Explanation:

The cardinality of the power set of a set with n elements is 2^n.

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