Algebra of Sets - The Fundamental Laws of Set Algebra

The Fundamental Laws of Set Algebra

The binary operations of set union and intersection satisfy many identities. Several of these identities or "laws" have well established names. Three pairs of laws, are stated, without proof, in the following proposition.

PROPOSITION 1: For any sets A, B, and C, the following identities hold:

commutative laws:

associative laws:

distributive laws:

Notice that the analogy between unions and intersections of sets, and addition and multiplication of numbers, is quite striking. Like addition and multiplication, the operations of union and intersection are commutative and associative, and intersection distributes over unions. However, unlike addition and multiplication, union also distributes over intersection.

The next proposition, states two additional pairs of laws involving three specials sets: the empty set, the universal set and the complement of a set.

PROPOSITION 2: For any subset A of universal set U, where Ø is the empty set, the following identities hold:

identity laws:

complement laws:

The identity laws (together with the commutative laws) say that, just like 0 and 1 for addition and multiplication, Ø and U are the identity elements for union and intersection, respectively.

Unlike addition and multiplication, union and intersection do not have inverse elements. However the complement laws give the fundamental properties of the somewhat inverse-like unary operation of set complementation.

The preceding five pairs of laws: the commutative, associative, distributive, identity and complement laws, can be said to encompass all of set algebra, in the sense that every valid proposition in the algebra of sets can be derived from them.

Note that if the complement laws are weakened to the rule, then this is exactly the algebra of propositional linear logic.