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Questions and Answers
What is the result of the operation A ∪ B if A = {5, 6} and B = {6, 7, 8}?
What is the result of the operation A ∪ B if A = {5, 6} and B = {6, 7, 8}?
How is the symmetric difference A ⊕ B defined?
How is the symmetric difference A ⊕ B defined?
What does the intersection A ∩ B yield when A = {2, 4, 6} and B = {1, 2, 3}?
What does the intersection A ∩ B yield when A = {2, 4, 6} and B = {1, 2, 3}?
What is the difference A − B when A = {3, 4, 5} and B = {4, 5, 6}?
What is the difference A − B when A = {3, 4, 5} and B = {4, 5, 6}?
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Which of the following represents the Cartesian product of A = {1, 2} and B = {x, y}?
Which of the following represents the Cartesian product of A = {1, 2} and B = {x, y}?
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If U = {1, 2, 3, 4, 5} and A = {2, 4}, what is the complement of A?
If U = {1, 2, 3, 4, 5} and A = {2, 4}, what is the complement of A?
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Given A = {a, b, c} and B = {b, c, d}, what is A ∩ B?
Given A = {a, b, c} and B = {b, c, d}, what is A ∩ B?
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What is the result of A ⊕ B if A = {1, 2, 3} and B = {2, 3, 4}?
What is the result of A ⊕ B if A = {1, 2, 3} and B = {2, 3, 4}?
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What is the correct definition of a set?
What is the correct definition of a set?
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Which of the following statements is true regarding the equality of sets?
Which of the following statements is true regarding the equality of sets?
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What is the cardinality of the set A = {a, b, c, d}?
What is the cardinality of the set A = {a, b, c, d}?
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Which set represents the power set of A = {a, b}?
Which set represents the power set of A = {a, b}?
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Which of the following defines an infinite set?
Which of the following defines an infinite set?
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What does the notation |A| represent?
What does the notation |A| represent?
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Which statement correctly represents the set of natural numbers?
Which statement correctly represents the set of natural numbers?
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What is a subset?
What is a subset?
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What is the result of the Cartesian product for sets A = {1, 2}, B = {a, b}, and C = {5, 6}?
What is the result of the Cartesian product for sets A = {1, 2}, B = {a, b}, and C = {5, 6}?
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Which of the following properties describes the relationship between the intersection and union of sets?
Which of the following properties describes the relationship between the intersection and union of sets?
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When are two ordered n-tuples considered equal?
When are two ordered n-tuples considered equal?
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What is the result of A × ∅ if A is a non-empty set?
What is the result of A × ∅ if A is a non-empty set?
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Which of the following is NOT a property of the empty set?
Which of the following is NOT a property of the empty set?
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According to De Morgan's laws, which statement is true?
According to De Morgan's laws, which statement is true?
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What does the theorem regarding the addition principle state for finite sets A and B?
What does the theorem regarding the addition principle state for finite sets A and B?
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Which of the following statements about the Cartesian product is true?
Which of the following statements about the Cartesian product is true?
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Study Notes
Discrete Structures - Sets
- A set is an unordered collection of distinct objects.
- Elements in a set are called members or elements.
- Order within a set is irrelevant. {a, b, c} = {c, b, a}
- Duplicates in a set do not change the set. {a, b, c, a} = {a, b, c}
- The set of natural numbers (N) is {1, 2, 3,...}
- The set of integers (Z) is {..., -3, -2, -1, 0, 1, 2, 3,...}
Sets and Subsets
- A subset of a set A is a set containing only elements of A.
- The power set of a set A (P(A)) is the set of all subsets of A.
- If A = {a, b}, then P(A) = {∅, {a}, {b}, {a, b}}
Operations on Sets
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Union (A ∪ B): The set containing all elements in A or B, or both.
- A ∪ B = {x | x ∈ A or x ∈ B}
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Intersection (A ∩ B): The set containing only elements common to both A and B.
- A ∩ B = {x | x ∈ A and x ∈ B}
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Difference (A – B): The set containing elements in A but not in B.
- A – B = {x | x ∈ A and x ∉ B}
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Complement (Ac): The set of elements in the universal set (U) but not in set A.
- Ac = U – A
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Symmetric Difference (A⊕B): The set of elements that are in either A or B, but not in both.
- A⊕B = {x | (x ∈ A and x ∉ B) or (x ∈ B and x ∉ A)}
Equality of Sets
- Two sets A and B are equal (A = B) if and only if they have the same elements.
- ∀x[x ∈ A ⇒ x ∈ B and x ∈ B ⇒ x ∈ A]
Cartesian Product
- The Cartesian product of two sets A and B (A × B) is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B.
- Example: A = {1, 2}, B = {a, b}. A × B = {(1, a), (1, b), (2, a), (2, b)}
Properties of Sets
- Commutative Properties: A ∪ B = B ∪ A and A ∩ B = B ∩ A
- Associative Properties: A ∪ (B ∪ C) = (A ∪ B) ∪ C and A ∩ (B ∩ C) = (A ∩ B) ∩ C
- Distributive Properties: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) and A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
- Properties of Complement: (Ac)c = A, A ∪ Ac = U, A ∩ Ac = ∅
- Properties of a Universal Set: A ∪ U = U, A ∩ U = A
- Properties of the Empty Set: A ∪ ∅ = A, A ∩ ∅ = ∅
Cardinality
- The number of distinct elements in a finite set is its cardinality (|A|)
Addition Principle
- If A and B are finite sets, then |A ∪ B| = |A| + |B|
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Description
Test your understanding of sets, subsets, and operations on sets in discrete structures. Explore concepts like union, intersection, and power sets with this quiz. Perfect for students learning about set theory in mathematics.
