Set Operations

Pre: Set Builder Notation, cardinality

Union

Definition

is the set of all elements belonging to either or

(see Set Builder Notation, logical operators)

Axiom - Set Union

(see If and Only If)

Properties

Commutativity:
Associativity:
Distributivity:

Additive Principle for the Union of Sets

Principle

For all additive set functions , and sets and (not necessarily disjoint), we have:

Proof

Recall the equation for finitely additivity. This only works for disjoint sets. For non-disjoint sets and , may not necessarily be empty set. The union operation however, removes any overlapping elements, so we must account for this removal of elements:

rearranging,

Intersection

Definition

is the set of all elements belonging to both and

Axiom - Set Intersection

Properties - Set Operators

Commutativity:
Associativity:
Distributivity:

Set Difference

Definition

(or ) is the set of all elements belonging to but not

Axiom - Set Difference

Complement

Definition - Without explicit mention of the universe

is the set of elements not in

Definition - With explicit mention of the universe

is the set of all elements in but not in

Equivalent to

Alternate notation: , ,

Axiom - Absolute Complement

Law

De Morgan's laws:

in fact, this is how De Morgan's Laws are originally defined, and the complement of a boolean is actually defined by sets too.

Generalized Set Functions

Definition

The union of multiple sets is written , where is a set of sets. The intersection of multiple sets is written .

Example

We can also do something like Summation and Product Notation, where we write:

to denote the start and end of .