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.
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 ฯต A, b ฯต B}, First element of all the ordered pair ฯต set A and the second element ฯต 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
Different types of relations:
Mainly relations are of three types: Reflexive, Symmetric and Transitive.
Reflexive Relation:
If (a, a) ฯต R(relation) ∀ a ฯต A, then the relation is called a reflexive relation.
Symmetric Relation:
If (a, b) ฯต R ⇒ (b, a) ฯต R, ∀ a ฯต A, then the relation is called a symmetric relation.
Transitive Relation :
If (a, b) ฯต R and (b, c) ฯต R, ⇒ (a, c) ฯต R, ∀ a ฯต A, then the relation is called transitive relation.
Equivalence Relation :
A relation which is reflexive, symmetric and transitive is called an equivalence relation.
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 ฯต A
One to one Function:
A function f : X → Y is defined to be one-one (or injective), if the images of distinct elements of X under f are distinct, i.e., for every x1 , x2 ∈ X, f(x1 ) = f(x2 ) implies x1 = x2 . Otherwise, f is called many-one.
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.
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.
Onto Function:
A function f: X → Y is said to be onto (or surjective), if every element of Y is the image of some element of X under f, i.e., for every y ∈ Y, there exists an element x in X such that f(x) = y
Bijective Functions (One-One and Onto)
A function which is one one and onto is called bijective function.
One-one and Onto Function:
A function f: X → Y is said to be one-one and onto (or bijective), if f is both one-one and onto.
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.
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.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)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 ฯต A and y ฯต B
iv) Solve f(x) = y to obtain x in terms of y
v) Replace x by f-1(y)
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
REFLECTION OF ACTIVITY
Students will be able to know the1) Cartesian product of two sets
2) Formation of different relations from the cartesian product.
3) Reflexive, Symmetric, Transitive and Equivalence relation.
4) Functions and their domain and range.
5) One-One, Many-One, Onto and Bijective Functions
IMMEDIATE FEEDBACK
After completing the above activities students will be able to
1) Identify different types of relations.
2) Identify different types of relations and their geometrical aspects.
3) Find the domain, range and codomain of different types of functions
CREATION (e.g. MIND-MAP, COLLAGE, GRAPH, MAP etc.)
SUBJECTS INTEGRATED
English : Write a brief note on Relations and Functions and correlate this knowledge with our day to day life activities'
Social Studies: Students may relate the relations and functions with their real life relations.
Drawing : Students can draw the free hand pictures of different types of functions or can also use some other material like thread, match sticks etc.
DIFFERENTIAL LEARNING
For Below Average Students
- Mind/ Concept maps
- Charts , Models and activity
- Simple questions
For Average Students
- Learning situations through watching video, creating collage, completing puzzles, assignment (click here)
For Above Average Students:
- Group Discussion
- Higher Order Thinking Skill questions
SKILLS ENHANCED
Observation skill, analytical skill, critical thinking, team work, constructive approach, interpersonal skill, engagement in learning process etc.
HOME ASSIGNMENTStudents will be given a worksheet for solving at home. Link of Google drive: link
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.
Competency based assessment can be taken so as to ensure if the learning outcomes have been achieved or not. e.g.- Puzzle
- Quiz
- Misconception check
- Peer check
- Students discussion
- Competency Based Assessment link: M C Q
Lot of thanks , sir Ji
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