#### 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**