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Questions and Answers
What is the intersection of two sets A and B denoted by?
Which property states that A ∩ B = B ∩ A?
If A = {1, 2, 3} and B = {2, 3, 4}, what is A ∩ B?
What is the intersection of a set with itself?
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What is represented by the region common to both circles in a Venn diagram?
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Study Notes
Intersection of Sets
Definition
- The intersection of two or more sets is a set containing all elements that are common to all the sets.
Notation
- The intersection of sets A and B is denoted by A ∩ B.
Properties
- Commutative Property: A ∩ B = B ∩ A
- Associative Property: (A ∩ B) ∩ C = A ∩ (B ∩ C)
- Distributive Property: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Examples
- If A = {1, 2, 3} and B = {2, 3, 4}, then A ∩ B = {2, 3}
- If A = {a, b, c} and B = {b, c, d}, then A ∩ B = {b, c}
Important Points
- The intersection of two sets can be empty if they do not have any common elements.
- The intersection of a set with itself is the set itself, i.e., A ∩ A = A.
- The intersection of a set with the universal set is the set itself, i.e., A ∩ U = A.
Venn Diagram Representation
- The intersection of two sets is represented by the region common to both circles in a Venn diagram.
Definition
- Intersection of sets consists of elements common to all sets involved.
Notation
- Denoted by A ∩ B for the intersection of sets A and B.
Properties
- Commutative Property: Order does not matter, A ∩ B = B ∩ A.
- Associative Property: Grouping does not affect the result, (A ∩ B) ∩ C = A ∩ (B ∩ C).
- Distributive Property: Combines intersection and union, A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
Examples
- For sets A = {1, 2, 3} and B = {2, 3, 4}, the intersection A ∩ B yields {2, 3}.
- For A = {a, b, c} and B = {b, c, d}, the intersection results in {b, c}.
Important Points
- The intersection can be empty if there are no common elements.
- Intersection of a set with itself is the set: A ∩ A = A.
- Intersection with the universal set provides the original set: A ∩ U = A.
Venn Diagram Representation
- In Venn diagrams, the intersection is visually represented by the overlapping region of the circles corresponding to the sets involved.
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Description
Understand the definition, notation, properties, and examples of the intersection of two or more sets in set theory. Learn about the commutative, associative, and distributive properties.
