Set theory is a field of mathematical logic that examines sets, which may be defined informally as collections of objects. In the 1870s, German mathematicians Richard Dedekind and Georg Cantor pioneered the current discipline of set theory. Set theory starts with a basic binary relationship between an object o and a set A.

Introduction to Sets

Set : Set is an unordered collection of objects. Members of the set are also called elements of the set.


Suggested Reading: Crisp Set – Complete Study


Typically, members of the set are represented using small letters and set itself is represented using a capital letter. If element a is member of set A, it is denoted by a ∈ A. If element ‘a’ is not a member of set A, it is denoted by a ∉ A.

Elements of a set are enclosed within the opening and closing curly braces and are separated by a comma. For example,

  • Set of alphabets : A = {a, b, c, . . . , z}
  • Set of digits : B = {0, 1, 2, . . . , 9}

Set Representation Methods

Listing method :

All the members of the set are exhaustively enumerated. Set of numbers greater than 100 is defined as, A = {101, 102, 103, 104, ….}.

This method does a listing of all elements. This is an inefficient way of writing big sets.

Describing properties:

Elements in set derived by stating the property of elements. Set of first five perfect square numbers is defined as, B = {1, 4, 9, 16, 25}.

Recursion method:

Elements are defined in terms of recursion, they are not explicitly listed as previous methods.

Set of first ten numbers is defined as C = { c | c ≤ 10 and c ∈ N}.

Recursion method is the best way of describing the set.

Types of Set

Subset

Set B is called a subset of set A if B is completely contained within A. In other words, if all members of B are also a member of A, then B is called a subset of A. And it is denoted by B ⊆ A.

Example:

If A = {1, 2, 3, 4, 5}, B = {1, 3, 5}, then B ⊆ A

Every set is a subset of itself. In the above example, A ⊆ A and B ⊆ B.

Superset :

Set A is called superset of set B if B is completely contained within A. In other words, if all the members of B are also a member of A, then A is called superset of A. And it is denoted by A ⊇ B.

Example:

If A = {1, 2, 3, 4, 5}, B = {1, 3, 5}, then A ⊇ B

Every set is a superset of itself. In the above example, A ⊇ A and B ⊇ B.

Proper subset :

If B ⊆ A and A ≠ B, then we say B is a proper subset of set A. This implies, B is a subset of A and A has at least one element which is not in B. It is denoted as B ⊂ A.

Example:

If A = {1, 2, 3, 4, 5}, B = {1, 2, 3, 4}, C = {1, 2, 3, 4, 5} holds following relations:

B ⊆ A, B ⊆ C, C ⊆ A

B ⊂ A,  B ⊂ C, but C ⊄ A

Proper superset :

If A ⊇ B and A ≠ B, then we say A is a proper superset of set A. This implies, A is a superset of B and A has at least one element which is not in B. It is denoted as A ⊃ B.

Example :

If A = {1, 2, 3, 4, 5}, B = {1, 2, 3, 4}, C = {1, 2, 3, 4, 5} holds following relations :

A ⊇ B, C ⊇ A, A ⊇ C,

A ⊃ B, but A is not a proper superset of C.

Empty set:

Set without any member is called an empty set. It is denoted as {  } or Φ.

Example :

If A = {1, 2, 3} and B = {5, 6} then intersection of these two sets is empty set.

Power Set

Power set P(A) of set A is a collection of all possible subsets of A.

Example :

If A = {a, b} then P(B) = {f, {a}, {b}, {a, b}}

If B = {1, 2, 3} then P(A) = { f, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }

If the size of the set is n, the size of its power set would be 2n.

Equal set:

Two sets A and B are said to be equal if A ⊆ B and B ⊆ A i.e. A = B. If each member of set A is also a member of B and converse is also true, we say that A and B are equal sets.

Example :

A = {1, 2, 3} and B = {1, 2, 3} are equal sets.

Operations on Sets

We can perform following operations on sets

Union:

Union of two sets A and B is defined as A ∪ B.

A ∪ B = { x : x ∈ A  or x ∈ B}.

If A = {1, 2, 3, 4} and B = {2, 4, 6, 8}, then A ∪ B = {1, 2, 3, 4, 6, 8}

In union, common elements appear only once.

Intersection:

Intersection is set of common elements from given two sets. Intersection of two sets A and B is defined as A ∩ B.

A ∩ B = { x : x ∈ A  and x ∈ B}. If A = {1, 2, 3, 4} and B = {2, 4, 6, 8}, then A ∩ B = {2, 4}

Difference:

Difference of two sets A and B is defined as A – B.

A – B = {x : x ∈ A  and x ∉ B}. If A = {1, 2, 3, 4} and B = {2, 4, 6, 8}, then A – B = {1, 3}, and B – A = {6, 8}

Complement:

Complement of set A with respect to some universal set U is defined as A’.

A’ = U – A = {x : x ∉ A }

Example :

If U = {1, 2, 3, 4, 5, 6} and A = {1, 4, 6} then  A’ = {2, 3, 5}

Cartesian product:

Cartesian product A ´ B of sets A and B is collection of all possible ordered pair (a, b) such that a Î A and b Î B.

A x B = {(a. b) : a ∈ A and b ∈ B}

Example :

If A = {1, 2, 3} and B = {x, y} then,

A x B = { (1, x), (2, x), (3, x), (1, y), (2, y), (3, y) }


Suggested Reading: Operations on crisp set


Venn diagram representation of sets

Sets and set operations are often represented using Venn diagram, which is a graphical representation of interaction between sets. Let us try to understand it with example. Assume the use case of some university having 100 students. To form the team for inter college sport tournaments, interests of the students were collected and we got following responses. Out of 100 students,

  • 60 students were interested in football only
  • 50 students were interested in cricket only
  • 45 students were interested in volleyball only
  • 30 students were interested in football and cricket both
  • 50 students were interested in volleyball and cricket both
  • 20 students were interested in football and volleyball both
  • 5 students were interested in all three games

This fact can be easily and elegantly represented by Venn diagram as shown below, where each circle corresponds to one set. Their overlapping shows the intersection of respective sets. Graphical representation is easy to understand and quick to analyze compared to its counter textual representation.

sett theory
Venn Diagram for sets

Thus, set theory provides a basic way of representing relations between objects or the elements and the set.


Additional Reading: Introduction to Set Theory [YouTube Video]