NDA 2019 Preparation: Maths Notes



A set is a well-defined collection of objects, known as elements or members of the set. Sets are usually denoted by capital letters & elements are usually denoted by small letters. If ‘a’ is an element of a set A, then we write a ϵ A (a belongs to A) otherwise a ∉ A (a doesn’t belong to A).

SET THEORY

Representation of sets: 

1. Tabular or Roster form : In this form, elements are listed within the pair of brackets { } and are separated by commas.
e.g.- N = {1, 2, 3, 4 ….} is a set of natural numbers 

2. Set-builder or Rule form: In this form, set is describe by a property that its member must satisfy.
e.g. - A = {x : x is natural number less than 10}

3. Statement form: In this representation, well defined description of the elements of the set is given. 
e.g. - The set of all even numbers less than 10. 

Different Types of sets: 
  • Null set : A set which does not contain any element is called a null set or an empty set or a void set. 
  • Singleton set: A set which contain only one element. 
  • Finite set : A set is called a finite set, if it is either void or its elements can be counted 
  • The number of distinct elements of a finite set A is called the cardinal number & it is denoted by n(A). 
  • Infinite set: A set which has unlimited number of elements is called infinite set. 
  • Equivalence sets: Two finite sets A and B are equivalent, if their cardinal numbers are same. 
  • Equal sets: Two sets are said to be equal if both have same elements. 
  • Note:– Equal sets are equivalent but equivalent sets may or may not be equal. 
  • Subset: If every element of set A is an element of set B, then A is called a subset of B sit is denoted by A ⊆ B.
  • Superset : If set B contains all elements of set A, then B is called superset of A & it is denoted by B ⊇ A.
  • Proper subset : A set A is said to be a proper subset of set B, if A is a subset of B & A is not equal to B. It is denoted by A ⊂ B.
  • Universal set : Universal set is a set which contains all objects, including itself. It is denoted by U. 
  • Power set : The set of all the possible subsets of A is called the power set & is denoted by P (A). 

Note:- 
  • The total number of subsets of a finite set containing n elements is 2ⁿ.
  • The total number of proper subsets of a finite set containing n elements is (2ⁿ –1).
  • If a set A has n elements, then its power set will contain 2ⁿ elements. 


Operations on sets: 
  • Union of two sets: The union of two sets A and B is the set of elements which are in A, in B or in both A & B. The union of A & B is denoted by A ∪ B. 
  • Intersection of two sets: The intersection of A & B is the set of all those elements which belong to both A & B & is denoted by A ∩ B. 
  • Disjoint of two sets: Two sets A & B are said to be disjoint if they don’t have any common element (i.e. A ∩ B = ϕ). 
  • Difference of two sets: The difference of sets A & B is the set of all those elements of A which do not belong to B. & is denoted by (A – B) or A\B.
  • Symmetric difference of two sets : The symmetric difference of sets A & B is the set (A – B) ∪ (B – A) and is denoted by A ∆ B. 
  • Complement of a set: The complement of a set A is the set of all those elements which are in universal set but not in A. It is denoted by  or U – A. 







                                  

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