A set is a well-defined, unordered collection of distinct objects. Each object that appears in this collection is called an element (or member) of the set.

Examples (using set enumeration):

  1. is a set that contains three elements.

Empty Set

The set contains no elements and is known as the empty set. We often use as the symbol.

Remark: Cardinality of : Number of elements in a finite set , denoted by Ex:

Set-builder Notation (Set Comprehension)

The idea is to define a set using a predicate; in particular, the set consists of all values that make the predicate true.

Set Operations


The union of two sets and , written , is the set of all elements belonging to either set or set (or both). Symbolically we write


The intersection of two sets and , written , is the set of all elements belonging to both set and set . Symbolically we write


The set-difference of two sets and , written (or ), is the set of all elements belonging to set but not . Symbolically we write


The complement of a set , written , is the set of all elements not in . Symbolically we write

Prove Subset

To prove that , prove the universally quantified implication:

Prove Equal Subsets

To prove that , we prove that and .



Types as sets: if

Set Comprehension

Empty Set

Universal Set The universal set consists of all the values of concern in any discussion (domain)

Set Equality


Proper Subset

Power Set The power set of a set is the set of all of its subsets. is the function that returns the power set of a set.


A singleton set is a set consisting only of one element.

Axioms for Set Functions

Set Union Set Intersection

Two sets, and , are disjoint if their intersection is empty, i.e. . Serendipity -> Disjoint Set Union

Absolute Complement is the universal set.

Set Difference Note that for set difference, the order of operands MATTERS, i.e.

For example, let , and .

Cartesian product (Set Product)

Also see Binary Relation for the list of other properties.

See Transformational Proof on Transformational Proof on Set Theory.