11th Class Maths: This note is designed by ‘Basics in Maths’ team. These notes to do help the CBSE 11^{th }class Maths students fall in love with mathematics and overcome their fear.

These notes cover all the topics covered in the CBSE 11^{th }class Maths syllabus and include plenty of formulae and concept to help you solve all the types of11^{th}Math problems asked in the CBSE board and entrance examinations.

## 1. SETS

**Well-defined objects:**

- All objects in a set must have the same general similarity or property.
- Must be able to confirm whether something belongs to the set or not.

** ****Set: – **A collection of well-defined objects is called a set.

∗ Sets are usually denoted by capital English alphabets like A, B, C, and so on.

∗ The elements in set are taken as small English alphabets like a, b, c, and so on.

∗ Set theory was developed by George canter.

• If any object belongs to a set, then it is called an object/element. We denote by ∈ to indicate that it belongs to. If it does not belong to the set then it is denoted by ∉.

Ex: – 1 ∈ N, 0 ∈ W, −1 ∈ Z, 0 ∉ N, etc.

**Methods of representing sets:**

**Roster or table or listed form: –**

In this form all the elements of the set are listed, and the elements are separated by commas and enclosed within braces { }.

Ex: – set of vowels in English alphabet = {a, e, I, o, u},

set of even natural numbers less than 10 = {2, 4, 6, 8} etc.

**Note: –** In roster form, an element is not repeated. We can list the elements in any order.

**Set builder form:**

Pointing an element in a set to x (or any symbols such as y, z, etc.) followed by a colon(:), next to write the properties or properties of the elements in that set and placed in flower brackets is called the set builder form.: Or / symbols read as ‘such that’

Ex: – {2, 4, 6, 8} = {x / x is an even and x ∈N, x< 10},

{a, e, i, o, u} = {x : x is a vowel in English alphabet}.

**Null set: – **(empty set or void set) the set which has no elements is called as a null set. It is denoted by ∅ or { }.

**Finite and infinite sets: – **If a set contains a finite no. of elements then it is called a finite set. If a set contains an infinite no. of elements then it is called an infinite set.

**Ex: – **A = {1, 2,3, 4} → finite set

** **B = {1, 2, 3, 4….}

** Equal sets: – **two sets A and B are said to be equal sets if they have the same elements., and write as A = B

** Ex: – **A = {1, 2, 3, 4}, B = {3, 1, 4, 2}

** ⟹** A = B.

**Subset: – **for any two sets A and B, if every element of set A is in set B, then we can say that A is a subset of B. It is denoted by A ⊂ B.

Ex: – If A = {1, 2, 3, 4, 5, 6, 7, 8}, subsets of A are {1}, {1, 3, 5}, {1,2,3,4}, and so on.

**Power set: – **set of all the subsets of a set A is called the power set of A. It is denoted by p(A).

Ex: – A = {1,2,3}

P(A) = {{1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1, 2, 3}, ∅}.

**Intervals:**

∗ Open interval: – (a, b) = {x: a< x <b} → set of rational numbers lies between a and b.

∗ Closed interval: – [a, b] = {x: a≤ x ≤b} → set of rational numbers lies between a and b, including a and b.

∗ Open – closed: – (a, b] = {x: a< x ≤b} → set of rational numbers lies between a and b, excluding a and including b.

∗ Closed-open: -[a, b) = {x: a≤ x <b} → set of rational numbers lies between a and b, including a and excluding b.

**Universal set: – **A set that contains all the subsets of it under our consideration is called a universal set. ** **

**Cardinal number of a set: –** Number of elements in a set A is called the cardinal number of that set A. It is denoted by n(A).

• If a set has n elements, then no. of elements of that set has 2^{n}

**Equivalent sets: – **two set A and B are said to be equivalent sets if n(A) = n(B) (they have the same cardinal number).

Ex: – A = {1, 2, 3}, B = {a, b, c}

n(A) = 3 and n(B) = 3

∴ A = B.

**Venn diagrams:**

U = {1, 2, 3, 4, 5, 6}

the relationship between sets is usually represented by means of diagrams which are known as ‘Venn diagrams. These diagrams consist of rectangles and circles. A universal set is represented by rectangles and subsets by circles.

U = {1, 2, 3, 4, 5, 6} A = {1, 2, 3} B = {1, 2}

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