 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.     