Set Theory: Intersection of 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.
• 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.