# 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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