Podcast
Questions and Answers
What is the set of all elements common to both sets A and B denoted by?
What is the set of all elements common to both sets A and B denoted by?
What is the commutative property of intersection?
What is the commutative property of intersection?
What is the result of A ∩ ∅, where ∅ is the empty set?
What is the result of A ∩ ∅, where ∅ is the empty set?
What is the result of A ∩ U, where U is the universal set?
What is the result of A ∩ U, where U is the universal set?
Signup and view all the answers
What is the set of all elements that belong to both sets A and B represented by in a Venn diagram?
What is the set of all elements that belong to both sets A and B represented by in a Venn diagram?
Signup and view all the answers
Study Notes
Intersection of Sets
- The intersection of two sets A and B, denoted by A ∩ B, is the set of all elements that are common to both sets.
- In other words, it is the set of elements that belong to both A and B.
Properties of Intersection
- Commutative property: A ∩ B = B ∩ A
- Associative property: (A ∩ B) ∩ C = A ∩ (B ∩ C)
- Distributive property: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
- Identity element: A ∩ U = A, where U is the universal set
- Zero element: A ∩ ∅ = ∅, where ∅ is the empty set
Examples
- If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then A ∩ B = {3, 4}
- If A = {a, b, c} and B = {d, e, f}, then A ∩ B = ∅, since there are no common elements
Venn Diagrams
- The intersection of two sets can be visualized using Venn diagrams, where the overlapping region represents the intersection of the sets.
Intersection of Sets
- The intersection of two sets A and B, denoted by A ∩ B, is the set of all elements that are common to both sets, meaning elements that belong to both A and B.
Properties of Intersection
- A ∩ B = B ∩ A, indicating the commutative property of intersection.
- (A ∩ B) ∩ C = A ∩ (B ∩ C), exhibiting the associative property of intersection.
- A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C), demonstrating the distributive property of intersection.
- A ∩ U = A, where U is the universal set, serving as the identity element for intersection.
- A ∩ ∅ = ∅, where ∅ is the empty set, acting as the zero element for intersection.
Examples
- If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then A ∩ B = {3, 4}, as these are the common elements in both sets.
- If A = {a, b, c} and B = {d, e, f}, then A ∩ B = ∅, since there are no common elements between the two sets.
Venn Diagrams
- Venn diagrams can be used to visualize the intersection of two sets, where the overlapping region represents the intersection of the sets, providing a graphical representation of the common elements.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Learn about the intersection of sets, its properties, and laws. Discover how to apply the commutative, associative, distributive properties, and more.
