appears in other texts; all three mean the same set.

Read that way, each law states that two different routes through these operations land on the same set.

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.

A ∩ (B ∪ C)UABC=(A ∩ B) ∪ (A ∩ C)UABC✓ Regions match — identity holds

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C), frozen

Both sides paint the same two lenses: the part of A that overlaps B, and the part of A that overlaps C. Intersection spreads across a union the way multiplication spreads across a sum. Step through the other three-set identities on the three-set laws explorer.

Union spreads across intersection as well, which has no counterpart in ordinary arithmetic.

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.

(A ∪ B)′UAB=A′ ∩ B′UAB✓ Regions match — identity holds

(A ∪ B)′ = A′ ∩ B′, frozen

Both sides paint the same region — everything outside both circles. The complement of a union keeps only what lies in neither set, which is exactly what is outside A and also outside B. Check the second De Morgan law, and every other two-set identity, on the two-set laws explorer.

The same exchange runs the other way: the complement of an intersection is the union of the complements.