Set Theory: Intersection of Sets
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Questions and Answers

What is the set of all elements common to both sets A and B denoted by?

  • A ∩ B (correct)
  • A - B
  • A ∪ B
  • B - A

What is the commutative property of intersection?

  • A - B = B - A
  • A ∩ B = B ∩ A (correct)
  • (A ∪ B) = (B ∪ A)
  • A ∪ B = B ∪ A

What is the result of A ∩ ∅, where ∅ is the empty set?

  • ∅ (correct)
  • U
  • A
  • B

What is the result of A ∩ U, where U is the universal set?

<p>A (C)</p> 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?

<p>The overlapping region (D)</p> 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.

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Learn about the intersection of sets, its properties, and laws. Discover how to apply the commutative, associative, distributive properties, and more.

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