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

Lesson Plan Math Class 12 (Ch-1) Relation & Functions


E- LESSON PLAN   SUBJECT MATHEMATICS    CLASS 10+2

Lesson Plan Class 12th Subject Mathematics for Mathematics Teacher. Effective way of Teaching Mathematics.Top planning by the teacher for effective teaching in the class.
lesson plan class 12

Board – CBSE

CLASS –XII

SUBJECT- MATHEMATICS

CHAPTER : 1  :-  Relations and Functions


TOPIC:-   Chapter : 1 : Relations and Functions

DURATION:-  

This chapter is divided into seven modules and is completed in fifteen class meetings.

PRE- REQUISITE KNOWLEDGE:-

TEACHING AIDS:- 

Green Board, Chalk,  Duster, Charts, smart board, projector, laptop etc.

METHODOLOGY:-   Lecture method

OBJECTIVES:-
  • Cartesian product and different types of relations.
  • Reflexive, symmetric, transitive and equivalence relations.
  • Different types of functions their domain and range.
  • One-one(injective) functions, onto (surjective)functions, bijective functions.
  • Composite functions.
  • Invertible and inverse of functions.
PROCEDURE :-
Start the session by asking the questions related to the set theory, Cartesian product of two sets, relations and functions their domains co-domains and range. Now introduce the topic relations and functions step by step as follows.

  TOPICS

COMPLETE EXPLANATION

 Module 1.  

Start the session with little description of the set theory, definition of Cartesian product relations and functions, their domain, co-domain and range.

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

Cartesian Product : 
If A and B are two non-empty sets, then the Cartesian product  A x B is defined as the set of all the ordered pairs of the elements from A to B such that  
A x B = {(a, b) : a ϵ A, b ϵ B},    First element of all the ordered pair ϵ set A and the second element ϵ set B

Relations: 
If A and B are two non-empty sets, then relation R from A to B is a subset of the Cartesian product A x B. This means that number of subsets is equal to the number of relations.

Domain of the relation: 
The set of all the first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R.

Range of the relation : 
The set of all second elements in a relation R from a set A to a set B is called the range of the relation R.

Co-domain of the relation: 
The whole set B is called the co-domain of the relation R.

Note: Range is the subset of the co-domain i.e.  Range  Co-domain

Function: 
A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. 


Note : A relation is said to be a function If no two ordered pairs have same first element. A function from A to B is denoted by f:A→B  

 Module 2.   Different types of relations: 

Mainly relations are of three types:  Reflexive, Symmetric and Transitive.

Reflexive Relation: 
If (a, a) ϵ R(relation)   a ϵ A, then the relation is called a reflexive relation.

Symmetric Relation: 
If (a, b) ϵ R  (b, a) ϵ R,    a ϵ A, then the relation is called a symmetric relation.

Transitive Relation : 
If (a, b) ϵ R and (b, c) ϵ R,  (a, c) ϵ R,   a ϵ A, then the relation is called transitive relation.

 Equivalence Relation : 
A relation which is reflexive, symmetric and transitive is called an equivalence relation.

 Module 3. 

Definitions of one-one, onto, many one and bijective functions and their arrow diagrams.

One-One Function(Injection): 
A function f from A to B is said to be one-one function or an injection if different elements of A have different images in B i.e. f(a) = f(b)  a = b for all a, b ϵ A

Onto Function(Surjection) : 
A function f from A to B is said to be onto function if every element of B is the image of some element in A or
If range of f = co-domain of f then function is called onto function.

Bijective Functions :
 A function which is one one and onto is called bijective function.

Many one Function: 
A function f from A to B is said to be many one function if two or more elements of set  A have  the same image in B.

 Module 4. 

Explain the Method of finding the domain and range of different functions.

Domain: Whole set A is called domain of the function. 
f:A
Range: 
Set of those elements of B which has pre - image in A is called range of the function.

 Module 5. Definition of  Composite functions:

Let f: A→B and g:B→C be two functions. Then the function gof: A→C defined by gof(x) =g(f(x)), for all x ∊ A
Composition is possible only if co-domain of first function is = domain of second function.

Now explain the method of finding fog and gof and related problems with the inverse of these functions.
Note:
The composition of function is not commutative i.e. fog ≠ gof
Composition of bijection is also a bijection.

 Module 6. Invertible function: 

A function which is one one and onto is called an invertible function.
If a function f is invertible then inverse of f   (f-1) exists.
Inverse of a function: Let f be a bijective function from A to B then a function g from B to A is called the inverse of f from A to B.
If f:A→B be a bijective function. Then a function g: B → A is called the inverse of f and is denoted by f-1(-1)
1(1)n
Algorithm
i) Obtain the function and check its bijectivity.
ii) If f is bijection then it is invertible.
iii) Put f(x) = y where x ϵ A and y ϵ B
iv) Solve f(x) = y to obtain x in terms of y
v) Replace x by f-1(y)

 Module 7. Properties of inverse of a function:

i) Inverse of a function is unique.
ii) Inverse of a bijection is also bijection.
iii) Inverse of composite function is also composite  or  (gof)-1 = f-1og-1


EXPECTED OUTCOMES:-
After studying this lesson students should know the 
  • concept Cartesian product, 
  • different types of relations, 
  • domain and range of relations, 
  • different types of functions their domain and range, 
  • bijection of functions and inverse of functions 
  • composition of functions.
STUDENTS DELIVERABLES:-
  • Review questions given by the teacher. 
  • Students should prepare the presentation in groups on different topics like Types of relations their domains and range. Different types of functions and method of finding their inverse. 
  • Students should solve NCERT problems with examples.
EXTENDED LEARNING:-
Students can extend their learning in Mathematics through the RESOURCE CENTRE.  Students can also find many interesting topics on mathematics at cbsemathematics.com

ASSESSMENT TECHNIQUES:-
  • Assignment sheet will be given as home work at the end of the topic. 
  • Separate sheets which will include questions of logical thinking and Higher order thinking skills will be given to the above average students.
  • Class Test , Oral Test , worksheet and Assignments. can be made the part of assessment.
  • Re-test(s) will be conducted on the basis of the performance of the students in the test.

THANKS FOR YOUR VISIT
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