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Questions and Answers
What is the result of the union of sets A = {1, 2, 3} and B = {3, 4, 5}?
What is the result of the union of sets A = {1, 2, 3} and B = {3, 4, 5}?
- {1, 2, 5}
- {1, 2, 3, 4, 5} (correct)
- {2, 3, 4}
- {3, 4}
If set A = {1, 2, 3, 4} and set B = {3, 4, 5, 6}, what is the complement of A with respect to the universal set U = {1, 2, 3, 4, 5, 6, 7}?
If set A = {1, 2, 3, 4} and set B = {3, 4, 5, 6}, what is the complement of A with respect to the universal set U = {1, 2, 3, 4, 5, 6, 7}?
- {1, 2}
- {5, 6}
- {4, 5, 6}
- {5, 6, 7} (correct)
Which of the following represents the intersection of sets A = {a, b, c} and B = {b, c, d}?
Which of the following represents the intersection of sets A = {a, b, c} and B = {b, c, d}?
- {c, d}
- {a}
- {a, d}
- {b, c} (correct)
Given sets A = {1, 2, 3} and B = {2, 3, 4}, what is the symmetric difference of A and B?
Given sets A = {1, 2, 3} and B = {2, 3, 4}, what is the symmetric difference of A and B?
What is the result of the symmetric difference A  B?
What is the result of the symmetric difference A  B?
Which of the following sets represents A - B if A = {x | x is even and less than 10} and B = {x | x is even and less than 6}?
Which of the following sets represents A - B if A = {x | x is even and less than 10} and B = {x | x is even and less than 6}?
According to the Addition Principle, how do you calculate the cardinality of the union of two finite sets A and B?
According to the Addition Principle, how do you calculate the cardinality of the union of two finite sets A and B?
What does De Morgan’s law state regarding the intersection and union of sets A and B?
What does De Morgan’s law state regarding the intersection and union of sets A and B?
What can be said about the empty set concerning the union operation?
What can be said about the empty set concerning the union operation?
Which property states that A ∪ B is equal to B ∪ A?
Which property states that A ∪ B is equal to B ∪ A?
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Study Notes
Operations on Sets
- An operation on a set combines two sets to produce a third set.
- Union of sets A and B is represented as A ∪ B and includes all elements in either A or B.
- Intersection of sets A and B is represented as A ∩ B and includes all elements common to both A and B.
- Disjoint Sets are sets with no elements in common, their intersection is an empty set.
- Operations like union and intersection can be performed on multiple sets at once.
- Complement of a set A, with respect to a universal set U, includes all elements in U that are not in A.
- Complement with respect to a set A, considers only elements belonging to A.
- Symmetric Difference includes elements belonging to either set A or B, but not both.
- Algebraic Properties of set operations include:
- Commutative: A ∪ B = B ∪ A and A ∩ B = B ∩ A
- Associative: A ∪ (B ∪ C) = (A ∪ B) ∪ C and A ∩ (B ∩ C) = (A ∩ B) ∩ C
- Distributive: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) and A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
- Idempotent: A ∪ A = A and A ∩ A = A
- Complement: (A')' = A, A ∪ A' = U, A ∩ A' = ∅, ∅' = U, and U' = ∅
- De Morgan's Laws: (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'
- Properties of Universal Set:
- A ∪ U = U
- A ∩ U = A
- Properties of Empty Set:
- A ∪ ∅ = A
- A ∩ ∅ = ∅
The Addition Principle
- The Addition Principle relates the cardinality of sets to the cardinality of their union.
- For finite sets A and B: |A ∪ B| = |A| + |B| - |A ∩ B|
- If sets A and B are disjoint, then |A ∪ B| = |A| + |B|.
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