Dictionary Rank of a Word | Permutations & Combinations

 PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

Set Theory Chapter 1 Class 11 Basics Concepts



Set Theory, Chapter1, Class-11
Basic concepts and formulas based on set theory chapter 1 cbse class 11. Useful and important points based on set theory. Revision notes of chapter 1 class 11 set theory.

Introduction Of Set Theory

The concept of sets serves as a fundamental part of the present day mathematics. Sets are used to define the concepts of relations and functions. The study of geometry sequences, probability etc. requires the knowledge of sets. The theory of sets developed by German Mathematician Georg Cantor(1845 - 1918)
Few examples of sets are given below
N : the set of all natural numbers.
W : the set of all whole numbers.
Z : the set of all integers.
Q : the set of all rational numbers.
R : the set of all real numbers.
Z+ : The set of positive integers.
Q+ : The set of positive rational numbers.
R+ : The set of positive real numbers.

Definition of set
collection of well defined and different  objects is called set. 
Sets are denoted by capital letters like A, B, C,……..etc.
Elements / objects / members of a set are represented by small letters like a, b, c, …….etc.
If a is the element of set A , then we say that a  A and is read as a belongs to set A. If a is not the element of A then we write a ∉ A and is read as a does not belongs to A.
The symbol "" is called epsilon and it means belongs to 

Method of representation of sets.

Method of representation of sets.
There are two methods of representing a set
a) Roster form or tabular form or Enumeration form :
In roster form all the elements are listed. The elements are being separated by commas and are enclosed within the braces {  }. For example the set of all vowels in English alphabet are written as  A = { a, e, i, o, u}.
While writing the elements in roster form the elements are not be repeated i.e. elements are taken as distinct.
Order of the elements of the set may be different. The above set may be written as  A = { o, e, a, i, u}
Note :Set of all real numbers is a set but cannot be written in the roster form.

b) Set-builder form or selector form or Rule method :  
In set builder form all the elements of a set possess a single common property, which is not possessed by any element out side the set.
For example the set of all vowels in English alphabet are written as
A = {x : x is a vowel in English alphabet}  and it is read as
Set A is the set of  all x such that x is the vowel of English alphabet.
Symbol " : " is named as colon and is read as such that.

Different types of sets

Empty set or Null set or Void set : 
A set which does not contain any element is called empty set or void set or null set. Empty set is denoted by ɸ
Finite set 
  • A set which is empty or consists of finite number of elements is called finite set. or
  • If the number of elements of a set are countable then it is called finite set. or 
  • A set is said to be finite if it has finite number of elements.
For Example:  A = {1, 2, 3} is a finite set
Infinite sets 
  • If the number of elements of a set are not countable then it is called infinite set. or 
  • A set is said to be infinite if it has infinite number of elements.
For Example:  A =  {1, 2, 3, …….} is an infinite set.
Equivalent sets:
Two finite sets A and B are said to be equivalent sets if number of element of both the sets A and B are equal.
For example : A = {1, 2, 3, 4, 5} and B = {a, b, c, d, e}.
Number of elements of set A and B are equal but their elements are different . Hence these sets are equivalent sets.
Equal Sets
Two sets are said to be equal if
a) Number of elements of both the sets are equal.
b) Each element of both the sets are same.
Eg. Two sets A = {1, 3, 5} and B = { 1, 1, 3, 5, 5} are said to be equal set because set B can be written as {1, 3, 5} because in set form repeated element can be written only once.
Order or Cardinal number of a finite set :
The number of different elements of a finite set A is called the order of a set. It is denoted by n(A) or O(A).
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Subsets and Power Sets

Set A is said to be a subset of set B if every element of set A is also the element of set B. It is denoted by A ⊂ B
A ⊂ B if  a  A ⇒ a  B
Super Set of A
If A is the subset of B then B is called the super set of A and is denoted by  B ⊃A
Proper Subset
A non-empty set A is said to be proper subset of B if A ⊂ B and A ≠ B
Φ and A are improper subsets of set A.
Comparable sets
Two sets A and B are said to be comparable sets if any one of them is the subset of other. If one set is not the subset of other then the sets are called non-comparable set.
Note
Every set is the subset of itself  
Empty set (ɸ)  is also the subset of every set.
N ⊂ W ⊂ Z ⊂ Q ⊂ R
Power Set: The collection of all the subsets of a set A is called the power set of A. It is denoted by P(A). Every element of P(A) is a set.
Example : Let set A = {1, 2, 3}
P(A) = {ɸ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
Total number of elements of P(A) = 8 = 23.
In general : If there are n elements in set A then total number of elements in the power set of A is  =  2n.
Universal Set : The basic set which is the super set for all the subsets used in a particular problem is called a Universal set. It is always denoted by U.

Different Types of  Intervals


Intervals are the subsets of Real number (R)
Open Interval 
The set of real numbers { x: a <  x < b and x ∊ R } is called an open interval and is denoted by (a, b).
All the points between a and b belongs to the open interval (a, b) but a, b themselves do not belong to this interval.

Close Interval
The interval which contains the end points also is called close interval and is denoted by [a, b], thus [a, b] = {x : a ≤ ≤ b and x ∊ R }
We can also write intervals open at one end and close at other end.

Semi-closed(semi- open) intervals
[a, b) = {x : a ≤ x < b and x ∊ R}
(a, b] = {x : a < x ≤ b and x ∊ R }
Graphical representations of ordered pairs
On the real number line different intervals described above are represented as shown below.

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Venn Diagrams

Most of the relationships between sets can be represented by means of diagrams which are known as Venn diagrams. In these diagrams universal set is denoted by rectangle and the subsets by circles. Elements of the subsets are written in their respective circles.


Union Of Sets

Let A and B are two sets. The union of A and B is the set which consists of all the elements of A and all the elements of B, the common elements being taken only once. The symbol “U” is used to denote union.
Symbolically we write  'A U B ' and is read as ‘A union B’.

Definition of Union of sets: 
The union of two sets A and B is the set which consists of all those elements which are either in A or in B. In symbols we can write
A U B = { x : x ∊ A or  x ∊ B }
In the venn diagram shaded portion represents  A U B
Properties of union of sets
i) It holds commutative law :  A U B = B U A       
ii) It holds Associative law      (A U B) U C  =  A U (B U C)
iii) A U ɸ = ɸ U A = A
iv) U ⋃ A = U

Intersection of sets

The intersection of sets A and B is the set of all elements which are common to both  A and B. The symbol ‘’ is used to denote the intersection.

Definition of intersection of sets:
The intersection of two sets A and B is the set of all those elements which belongs to both A and B. Symbolically we write
             A ∩ B = { x : x ∊ A and  x ∊ B }
Properties of intersection
i) It holds commutative law :  A ∩ B = B ∩ A
ii) It holds Associative law :     A ∩ (B ∩ C) = (A ∩ B) ∩ C 
iii) A ∩ ɸ = ɸ ∩ A = ɸ
vi) It holds the distributive law :   ∩ (B U C) = (A ∩ B) U (A ∩ C) 
v) U ∩ A = A
Difference of sets:
The difference of two sets A and B in this order is the set of elements which belongs to A but not to B. Symbolically, we write A – B and read as “ A minus B” Symbolically we can write
               A – B = { x : x ∊A and x ∉ B}
                  B – A = { x : x ∊B and x ∉ A}
Disjoint sets: 
Two sets A and B are said to be disjoint if their intersection is ‘ɸ’. Or
If A ∩ B = ɸ, then A and B are said to be disjoint sets.

Symmetric Difference of two sets:
If A and B are two sets, then the set (A - B) U (B - A) is called symmetric difference of A and B and is denoted by A Δ B.

A Δ B = {x : (x ∊ A and x ∉ B) or (x ∊ B and x ∉ A)}

Complement of a set

Let U be a universal set and A is a subset of U. Then the complement of A is the set of all elements of U which are not the elements of A. complement of A is denoted by A' and is given by
      A' = {x :  x ∊ U and x ∉ A}
In the following venn diagram shaded portion denotes the complement of A (or A')

Properties of complement set
i) A U A' = U        ii) A ∩ A' = ɸ    iii) (A')' = A
iv) ɸ' = U       v)  U' = ɸ
De-Morgan’s Law  
i) (A U B)' = A'  B'    (ii)  (A ∩ B)' = A' U  B'   
Formulas used in practical problems
i) n(A U B) = n(A) + n(B) – n(A  B)
where n represents the number of element in a set.
ii)  n(A U B U C) = 
           n(A) + n(B) + n(C) – n(AB) – n(B ∩ C) – n(C ∩ A) + n(A ∩ ∩ C)

If A and B are two sets then
iii) n(A only) = n(A) – n(A ∩ B) = n(A - B)
iv) n(B only) = n(B) - n(∩ B) = n(B - A)
These formulas are also be useful for finding the probabilities ( Chapter 16)
Symbols
Meaning
U (Union)
or
∩ (Intersection)
and
- (Minus)
 and not
 A' (complement)
not A 
At least
U (Union)
Either or
U (Union)
Neither nor
∩ (Intersection)

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