Set Theory Rules

Idempotent Laws

Associative Laws

Commutative Laws

Distributive Laws

Complement Laws

De Morgan’s Laws

Idempotent Laws

	law	formula	explanation
	Idempotent Law for Union	$A \cup A = A$	A set unioned with itself remains the same set.
	Idempotent Law for Intersection	$A \cap A = A$	A set intersected with itself remains the same set.

Associative Laws

	law	formula	explanation
	Associative Law for Union	$(A \cup B) \cup C = A \cup (B \cup C)$	Grouping of sets under union does not affect the result.
	Associative Law for Intersection	$(A \cap B) \cap C = A \cap (B \cap C)$	Grouping of sets under intersection does not affect the result.

Commutative Laws

	law	formula	explanation
	Commutative Law for Union	$A \cup B = B \cup A$	The order of sets in a union does not affect the result.
	Commutative Law for Intersection	$A \cap B = B \cap A$	The order of sets in an intersection does not affect the result.

Distributive Laws

	law	formula	explanation
	Distributive Law of Union over Intersection	$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$	Union distributes over intersection.
	Distributive Law of Intersection over Union	$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$	Intersection distributes over union.

Identity Laws

	law	formula	explanation
	Identity Law for Union	$A \cup \emptyset = A$	The union of a set with the empty set is the set itself.
	Identity Law for Intersection	$A \cap U = A$	The intersection of a set with the universal set is the set itself.
	Annihilation Law for Intersection	$A \cap \emptyset = \emptyset$	The intersection of a set with the empty set is the empty set.
	Annihilation Law for Union	$A \cup U = U$	The union of a set with the universal set is the universal set.

Complement Laws

	law	formula	explanation
	Complement Law for Union	$A \cup A^c = U$	A set unioned with its complement gives the universal set.
	Complement Law for Intersection	$A \cap A^c = \emptyset$	A set intersected with its complement gives the empty set.
	Complement of Universal Set	$U^c = \emptyset$	The complement of the universal set is the empty set.
	Complement of Empty Set	$\emptyset^c = U$	The complement of the empty set is the universal set.
	Double Complement Law (Involution)	$(A^c)^c = A$	Taking the complement twice returns the original set.

De Morgan’s Laws

	law	formula	explanation
	De Morgan’s Law for Union	$(A \cup B)^c = A^c \cap B^c$	The complement of a union is the intersection of the complements.
	De Morgan’s Law for Intersection	$(A \cap B)^c = A^c \cup B^c$	The complement of an intersection is the union of the complements.