Set laws

Set laws describe the fundamental rules of how set operations work. These laws often correspond to familiar rules from algebra and logic and give us tools to simplify and manipulate expressions with sets.

 

 

Identity laws

The identity laws describe how union and intersection work with the empty set and the universe \( \large U\):

 

$$ \large A \cup \emptyset = A $$

$$ \large A \cap U = A $$

 

These laws show that the empty set does not add anything to the union, and that the universe does not remove anything from the intersection.

 

 

Commutative, associative and distributive laws

These laws show that the order and grouping of operations do not change the result:

 

• Commutativity:

$$ \large A \cup B = B \cup A, \quad A \cap B = B \cap A $$

• Associativity:

$$ \large (A \cup B) \cup C = A \cup (B \cup C) $$

$$ \large (A \cap B) \cap C = A \cap (B \cap C) $$

• Distributivity:

$$ \large A \cup (B \cap C) = (A \cup B) \cap (A \cup C) $$

$$ \large A \cap (B \cup C) = (A \cap B) \cup (A \cap C) $$

 

 

De Morgan's laws

De Morgan's laws connect union and intersection with complement:

 

$$ \large (A \cup B)^c = A^c \cap B^c $$

$$ \large (A \cap B)^c = A^c \cup B^c $$

 

These laws are important in both set theory and logic.

 

 

Absorption laws

The absorption laws describe how a set combined with an operation on itself and another set simplifies:

 

$$ \large A \cup (A \cap B) = A $$

$$ \large A \cap (A \cup B) = A $$

 

 

Generalization to more sets

Many set laws can be extended to more than two sets. For example:

 

$$ \large A \cup (B \cup C \cup D) = (A \cup B) \cup (C \cup D) $$

$$ \large A \cap (B \cap C \cap D) = (A \cap B) \cap (C \cap D) $$

 

This generalization shows that most laws apply not only to two sets, but can be extended to any number.