# SET theory

#### Definition:

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 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 . Here B is called superset of A and also written as • 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 and implies .
• If A is a set with , then the number of subsets of A are .
• 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) = { , {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 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 (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., or • Intersection of Two Sets:   The intersection of two sets A and B, written as (read as ‘A intersection B’), is the set consisting of all the elements which are both in A and  B. i.e., and • 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. and • Symmetric difference of Two Sets:   The symmetric difference of two sets  A and B, is
• • Complement of a Sets:   The complement of a set A, written as or , is the set consisting of all the elements which are not in A. i.e., and #### Algebra of Set operation

• Idempotent Law:  • Identity Law:  For any set A,  • Commutative Law:  • Associative  Law:   For any three sets A, B, C  • Distributive  Law:   For any three sets A, B, C  • De-morgan’s  Law:   For any two sets A and 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 .   •

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