1. Naive Set Theory

A set is a collection of distinct objects called elements. A set exists whenever we can decide whether a given object belongs to it. This informal approach — where any definable collection is considered a set — is what makes it "naive."

Basic Notation

  • — is an element of (or belongs to ).
  • — is not an element of .
  • — is a subset of (every element of is also in ).
  • — is a proper subset of ( but ).

Two sets are equal if they have exactly the same elements, which is the same as saying and .

Set Specifications

There are two common ways to describe a set:

  • Roster form: Simply list the elements inside curly braces. Example: . Order and repetition don't matter, so .
  • Set-builder notation: Describe the property that determines membership. Example: .

Special Sets

  • Empty set ( or ): The set with no elements. It is a subset of every set.
  • Universal set (): The set of all objects under consideration in a given context.
  • Singleton: A set with exactly one element, e.g. .

Basic Operations

Operation Notation What it gives you
Union Elements in or (or both)
Intersection Elements in both and
Relative complement Elements in but not in
Absolute complement or Elements not in (relative to )

Two sets are disjoint if — they share nothing.

Symmetric Difference

The symmetric difference of and is the set of elements that belong to exactly one of the two sets:

If you think of union as "or" and intersection as "and", symmetric difference is like an exclusive or (XOR) — an element is in if it is in or in , but not in both.

Venn Diagrams

Venn diagrams represent sets as overlapping circles inside a rectangle (the universal set). Shaded regions show the result of set operations. Click the buttons below to see each operation.

A B U

Power Sets

The power set of , denoted , is the set of all possible subsets of :

If has elements, then has elements. For example:

The empty set and the original set itself are always included.

Set Identities

These are algebraic rules that sets follow, much like the rules of arithmetic:

Law Formula
Commutative ,
Associative ,
Distributive ,
Idempotent ,
De Morgan's ,
Complement , ,

A useful way to check these is to draw a Venn diagram — the shaded regions will match on both sides of the equation.

Cartesian Products

The Cartesian product of and is the set of all ordered pairs you can form by taking one element from each:

Key points:

  • Order matters: unless .
  • Ordered -tuples: extend the idea to more than two sets.
  • Coordinate geometry: gives us the familiar -plane. Every point in the plane is an ordered pair of real numbers.
  • Cardinality: If and , then .

Next: ZFC Axioms