Set theory

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Basic Definition[edit]

The term "set" can be thought as a well-defined collection of objects. In set theory, These objects are often called "elements".

  • We usually use capital letters for the sets, and lowercase letters for the elements.
  • If an element belongs to a set , we can say that " is a member of the set ", or that " is in ", or simply write .
  • Similarly, if is not in , we would write .

(Example) . In this case, is the element and is the set of all real numbers.


Example of a common notation style for the definition of a set:

  • is a set
  • is a property (The elements of may or may not satisfy this property)
  • Set can be defined by writing...

This would read as "the set of all in ,such that of ."


There are two ways that we could show which elements are members of a set: by listing all the elements, or by specifying a rule which leaves no room for misinterpretation. In both ways we will use curly braces to enclose the elements of a set. Say we have a set that contains all the positive integers that are smaller than ten. In this case we would write . We could also use a rule to show the elements of this set, as in .

In a set, the order of the elements is irrelevant, as is the possibility of duplicate elements. For example, we write to denote a set containing the numbers 1, 2 and 3. Or,.


A is a subset of B.

Formal universal conditional statement: "set A is a subset of a set B"

  • , if , then


  • such that and
If and only if: 
    for all :
        If: ( is an element of A)
        then: ( is an element of B)
    set A is a subset of set B

Truth Table Example:

if , then and
1 1 1 0
1 0 0 1
0 1 1 0
0 0 1 0

A proper subset of a set is a subset that is not equal to its containing set. Thus

A is a proper subset of B

Set Identities[edit]

Let all sets referred to below be subsets of a universal set U.

(a) A ∪ ∅ = A and (b) A ∩ U = A.

5. Complement Laws:

(a) A ∪ A c = U and (b) A ∩ A c = ∅.

6. Double Complement Law:

(A c ) c = A.

7. Idempotent Laws:

(a) A ∪ A = A and (b) A ∩ A = A.

8. Universal Bound Laws:

(a) A ∪ U = U and

(b) A ∩ ∅ = ∅.

Identity For all sets A
Identity Laws:

Complement Laws:

Double Complement Law:
Idempotent Laws:

Universal Bound Laws:
Identity For all sets A and B
Commutative Laws:

2. Associative Laws: For all sets A, B, and C,

(a) (A ∪ B) ∪ C = A ∪ (B ∪ C) and

(b) (A ∩ B) ∩ C = A ∩ (B ∩ C).

3. Distributive Laws: For all sets, A, B, and C,

(a) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and

(b) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

For all sets A and B,

De Morgan’s Laws:

(a) (A ∪ B) c = A c ∩ B c and (b) (A ∩ B) c = A c ∪ B c .

Absorption Laws:

(a) A ∪ (A ∩ B) = A and (b) A ∩ (A ∪ B) = A.

Set Difference Law:

  • A − B = A ∩ B c .

Complements of U and ∅:


The cardinality of a set is the number of elements in the set. The cardinality of a set is denoted

Types of Sets by Cardinality[edit]

A set can be classified as finite, countable, or uncountable.

  • Finite Sets are sets that have finitely many elements, is a finite set of cardinality 3.
  • Countable Sets are sets that have as many elements as the set of natural numbers, so the set of rational numbers is countable.
  • Uncountable Sets are sets that have more elements than the set of natural numbers, so the set of real numbers is uncountable.

Common Sets of Numbers[edit]

Diagram to demonstrate the number systems ℝ, ℚ, ℤ and ℕ as sub-sets of each other.
  • is the set of Naturals
  • is the set of Integers
  • is the set of Rationals
  • is the set of Complex Numbers

Partitions of Sets[edit]

In many applications of set theory, sets are divided up into non-overlapping (or disjoint) pieces. Such a division is called a partition.

Two sets are called disjoint if, and only if, they have no elements in common.

A and B are disjoint ⇔ A ∩ B = ∅.

Sets are mutually disjoint (or pairwise disjoint or nonoverlapping)

if, and only if, no two sets and with distinct subscripts have any elements in

common. More precisely, for all

whenever .

Power Sets[edit]

The power set of a set A is all possible subsets of A, including A itself and the empty set. Which can be represented:

For the set

Learning resources[edit]