Set theory

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Basic DefinitionEdit

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 IdentitiesEdit

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 CardinalityEdit

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 NumbersEdit

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 SetsEdit

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 SetsEdit

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 resourcesEdit