Set theory operations follow several identities and laws that govern how sets behave under union, intersection, and complement. These laws are foundational in mathematics, computer science, and logic.
\( A \cup A = A \): Union with itself gives the same set
\( A \cap A = A \): Intersection with itself gives the same set
\( A \cup B = B \cup A \): Union is commutative
\( A \cap B = B \cap A \): Intersection is commutative
\( (A \cup B) \cup C = A \cup (B \cup C) \): Grouping doesn’t affect union
\( (A \cap B) \cap C = A \cap (B \cap C) \): Grouping doesn’t affect intersection
\( A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \)
\( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \)
These distribute one operation over another
\( U' = \varnothing \): Complement of universal set is empty
\( \varnothing' = U \): Complement of empty set is universal
\( A \cup U = U \): Anything united with universal set is universal
\( A \cap U = A \): Intersection with universal set gives original set
\( A \cup \varnothing = A \): Union with empty set gives original set
\( A \cap \varnothing = \varnothing \): Intersection with empty set gives empty set
\( (A')' = A \): Double complement restores the set
\( A \cup A' = U \): Union of a set and its complement is universal
\( A \cap A' = \varnothing \): Intersection of a set and its complement is empty
\( (A \cup B)' = A' \cap B' \): Complement of union is intersection of complements
\( (A \cap B)' = A' \cup B' \): Complement of intersection is union of complements