Set Law

A Set Law a Constraint in a Set System.

• AKA: Set Law-Set Axiom.
• Context:
  • It can be defined in terms of Set Operations.
• Example(s):
  • Commutative Set Law.
  • Associative Set Law.
  • Distributive Set Law.
  • Identity Set Law.
  • Complement Set Law.
• See: Axiom Addition, Multiplication, Associativity, Commutativity, Reflexive Relation, Antisymmetric Relation, Transitive Relation, Set (Mathematics), Naive Set Theory, Axiom.

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References

2019

• (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Algebra_of_sets#The_fundamental_properties_of_set_algebra Retrieved:2019-11-10.
  • The binary operations of set union ([math]\displaystyle{ \cup }[/math] ) and intersection ([math]\displaystyle{ \cap }[/math] ) satisfy many identities. Several of these identities or "laws" have well established names.

Commutative property:

• [math]\displaystyle{ A \cup B = B \cup A }[/math]
• [math]\displaystyle{ A \cap B = B \cap A }[/math]

Associative property:

• [math]\displaystyle{ (A \cup B) \cup C = A \cup (B \cup C) }[/math]
• [math]\displaystyle{ (A \cap B) \cap C = A \cap (B \cap C) }[/math]

Distributive property:

• [math]\displaystyle{ A \cup (B \cap C) = (A \cup B) \cap (A \cup C) }[/math]
• [math]\displaystyle{ A \cap (B \cup C) = (A \cap B) \cup (A \cap C) }[/math]

The union and intersection of sets may be seen as analogous to the addition and multiplication of numbers. Like addition and multiplication, the operations of union and intersection are commutative and associative, and intersection distributes over union. However, unlike addition and multiplication, union also distributes over intersection.

Two additional pairs of properties involve the special sets called the empty set Ø and the universe set [math]\displaystyle{ U }[/math] ; together with the complement operator ([math]\displaystyle{ A^C }[/math] denotes the complement of [math]\displaystyle{ A }[/math] . This can also be written as [math]\displaystyle{ A' }[/math] , read as A prime). The empty set has no members, and the universe set has all possible members (in a particular context).

Identity :

• [math]\displaystyle{ A \cup \varnothing = A }[/math]
• [math]\displaystyle{ A \cap U = A }[/math]

Complement :

• [math]\displaystyle{ A \cup A^C = U }[/math]
• [math]\displaystyle{ A \cap A^C = \varnothing }[/math]

The identity expressions (together with the commutative expressions) 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 formulae—the commutative, associative, distributive, identity and complement formulae—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 formulae are weakened to the rule [math]\displaystyle{ (A^C)^C = A }[/math], then this is exactly the algebra of propositional linear logic.

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