# Sets Formula

Master Sets Formula effortlessly with our comprehensive resource! Access chapter-wise formulas specifically designed for Class 11 and Madhyamik students in the field of Mathematics on our website. Strengthen your mathematical skills and excel in your exams with our user-friendly Sets Formula guide.

## Introduction

Set: A set is a collection of well-defined objects which are distinct from each other. Sets are usually denoted by capital letters A, B, C, … and elements are usually denoted by small letters a, b, c, …

If a is an element of a set A, then we write a ∈ A.

## Standard Notations

 Symbols Meaning N A set of all-natural numbers. W A set of all whole numbers. Z A set of all integers. Z+/Z– A set of all positive/negative integers. Q A set of all rational numbers. Q+/Q– A set of all positive/negative rational numbers. R A set of all Real Numbers. R+/R– A set of all positive/negative real numbers. C A set of all complex numbers.

## Representation of sets

1. Roster form/Tabular form: In this method, a set is described by listing the elements, separated by commas and enclosed within braces. Example – A = {a, e, i, o, u}
2. Set Builder Form: In this method, we write down a property or rule which gives us all the elements of the set. Example – A = {x : x is a vowel in English alphabet}

## Types of sets

1. Empty set: A set containing no element, it is denoted by ∅ or {}.
2. Singleton set: A set containing a single element.
3. Finite Set: A set containing a finite number of elements or no element.
4. Infinite set: A set containing an infinite number of sets.
5. Equal set: Two sets A and B are said to be equal if every element of A is a member of B and every element of B is a member of A and we write it as A = B.

### Subset and Superset

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 B is called a superset of A and written as A ⊆ B or B ⊇ A.

### Power Set

The set formed by all the subsets of a given set A, is called a power set of A, denoted by P(A).

### Universal Set (U)

A set consisting of all possible elements which occur under consideration is called a universal set.

### Proper Subset

If A is a subset of B and A ≠ B, then A is called a proper subset of B and we write it as A ⊂ B.

### Disjoint Sets

Two sets A and B are called disjoint if A ∩ B = ∅ i.e. they do not have any common element.

In a Venn diagram, the universal set is represented by a rectangular region and its subset is represented by circle or a closed geometrical figure inside the rectangular region.

## Operation on Sets

### Union of sets

The union of two sets A and B, denoted by A ∪ B, is the set of all those elements which are either in A or in B or both in A and B.

### Intersection of sets

The intersection of two sets A and B, denoted by A ∩ B, is the set of all those elements which are common in both A and B.

### Difference of sets

For two sets A and B, the difference A – B is the set of all those elements of A which do not belongs to B.

### Complement of a Set

If A is a set with U as a universal set, then the complement of a set A, denoted by A’ or Ac is the set U – A.

## Formulae on Number of Elements in Sets

(i) n(A ∪ B) = n (A) + n(B) – n(A∩B)

(ii) n(A ∪ B) = n (A) + n(B), if A and B are disjoint sets.

(iii) n(A – B) = n(A) – n(A∩B)

(iv) n(B – A) = n(B) – n(A∩B)

(v) n(A ∪ B ∪ C) = n (A) + n(B) + n(c) – n(A∩B) – n(B∩C) – n(A∩C) + n(A∩B∩C)

(vi) n(A’∪B’) = n(A ∩ B)’

(vii)  n(A’∩B’) = n(A ∪ B)’

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