SET theory


A Set is a collection of well-defined objects or elements. The term well-defined stands for definition assigns it a unique interpretation or value.

For example the collection of all the vowels in English alphabets is a set. where as the collection of Tall tree is not a set. As the term “Tall” is not well defined.

Representation of a Set:

There are two methods of representation of a set:
(1) Roster or Tabular form: In this method we describe a set by listing the elements, separated by commas within curly brackets.
(2) Set builder form: Here all the elements of a set possess a single common property which is not possess by any other elements out side the set.

Types of Sets:

  • Empty Set: A Set which does not contains any elements is called the empty set or the null set or the void set. The empty set is denoted by the symbol \phi or \{\}
  • Finite and Infinite Set: A Set which is empty or consists of a definite number of elements is called finite set, otherwise the set is called infinite set.
  • Singleton Set: A set consists of only one element.
  • Subset: Let A and B be two sets. If every element of A is an element of B, then A is called a subset of B and we write A \subseteq B. Here B is called superset of A and also written as B \supseteq A
    • Every set is a subset of itself.
    • The empty set is the subset of every set.
  • Equal Sets: Two sets A and B are said to be equal if every element of A are also an element of B and every element of the set B are also in A. In other words A \subseteq B and B \subseteq A implies A=B.
  • If A is a set with n (A) = m, then the number of subsets of A are 2^m.
  • Power Set: The set of all subsets of a given set A is called the power set of A and is denoted by P(A).
    • For example, if A = {1, 2, 3}, then P(A) = {\phi, {1}, {2}, {3}, {1, 2} {1, 3}, {2, 3}, {1, 2, 3}}
    • if A has n elements, then its power set P(A) contains exactly 2^n elements
  • Universal Set: A Set U that contains all sets in a given context is called the universal set.

Operations on Sets

  • Union of Two Sets:   The union of two sets A and B, written as A \cup B  (read as ‘A union B’), is the set consisting of all the elements which are either in A or in B or in both. i.e.,   A \cup B=\{x: x \in A or x  \in B\}
  • Intersection of Two Sets:   The intersection of two sets A and B, written as A \cap B  (read as ‘A intersection B’), is the set consisting of all the elements which are both in A and  B. i.e.,   A \cap B=\{x: x \in A  and x \in B\}
  • Difference of sets:   The difference of two sets A and B is the set of all the elements of A which do not belongs to B.  A - B=\{x: x \in A and  x \notin B\}
  • Symmetric difference of Two Sets:   The symmetric difference of two sets  A and B, is
      • A \Delta B= (A \cup B) - (A \cap B)
      • A \Delta B= (A - B) \cup (B-A)  

Complement of a Sets:   The complement of a set A, written as A' or A^C  , is the set consisting of all the elements which are not in A. i.e.,   A '=\{x: x \in U and x \notin A\}

Algebra of Set operation

  • Idempotent Law:  A \cup A=A  A \cap A=A
  • Identity Law:  For any set A,  A \cup \phi=A  A \cap U=A
  • Commutative Law:   A \cup B=B \cup A  A \cap B=B \cap A
  • Associative  Law:   For any three sets A, B, C
    A \cup (B \cup C)= (A \cup B) \cup C
    A \cap (B \cap C)= (A \cap B) \cap C
  • Distributive  Law:   For any three sets A, B, C
    A \cup (B \cap C)= (A \cup B) \cap ((A \cup C)
    A \cap (B \cup C)= (A \cap B) \cup ((A \cap C)
  • De-morgan’s  Law:   For any two sets A and B
    (A \cup B)'= A' \cap B'
    (A \cap B)'= A' \cup B'
  • Cardinality or Order of a finite set:   The cardinality of a finite set is the number of elements in the set. The cardinality is denoted by |A|.
    |A \cup B|=|A|+|B|- |A \cap B|
    (|A \cup B\cup C|=|A|+|B|+|C|- |A \cap B|- |B \cap C|- |A \cap C|+-|A \cap B\cap C|
    |A \Delta B|=|A-B|+|B-A|=|A|+|B|-2 |A \cap B|

For more understanding of the concept you can view some of the solved problems and MCQs regarding Set theory