Union (∪)
The union of two sets A and B contains all elements that are in A, or in B, or in both.
A ∪ B = {x | x ∈ A or x ∈ B}
Example: {1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}Intersection (∩)
The intersection of two sets A and B contains all elements that are in both A and B.
A ∩ B = {x | x ∈ A and x ∈ B}
Example: {1, 2, 3} ∩ {2, 3, 4} = {2, 3}
Two sets are disjoint if A ∩ B = ∅Difference and Complement
Difference: A − B = {x | x ∈ A and x ∉ B}
Example: {1, 2, 3} − {2, 3, 4} = {1}
Complement: Aᶜ = U − A = {x | x ∈ U and x ∉ A}
Example: If U = {1,2,3,4,5} and A = {2,4}, then Aᶜ = {1,3,5}Set Identities
A ∪ A = A A ∩ A = A
A ∪ ∅ = A A ∩ U = A
A ∪ Aᶜ = U A ∩ Aᶜ = ∅
(A ∪ B)ᶜ = Aᶜ ∩ Bᶜ (De Morgan)
(A ∩ B)ᶜ = Aᶜ ∪ Bᶜ (De Morgan)
A − B = A ∩ Bᶜ