Class 12 Mathematics Chapter 1 Notes
Mathematics is regarded to be one of the most demanding subjects for students of all classes. The students of higher classes face difficulty preparing for their examinations since high-level Mathematics concepts are introduced. The Class 12 Mathematics Chapter 1 notes- Relations and Functions is an important chapter and defines different concepts along with their graphs.
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Key Topics Covered In Class 12 Mathematics Chapter 1 Notes
The key topics covered under class 12 Mathematics chapter 1 notes include the following.
Relation:
Definition: Relation is the connection between two or more sets of values. If (a,b) ∈ R, then a R b means a is related to b under R.
For example: if A = {9, 16, 25} and B = {5, 4, 3, -3, -4, -5} then set A is related to set B under R, where R = {(x,y), x=y2 / x A and y B}
Types of Relations:
The types of relation mentioned in class 12 Mathematics chapter 1 notes are
The relation R in any set A is said to be empty if no element in A is related to any element of the same set, i.e., R = φ A × A
Consider set A= {2, 3, 5} and R= {(x,y), x + y > 9}, then R is an empty relation or void relation.
The relation R in any set A is said to be universal if every element in A is related to every element of the same set, R = A × A
For set A= {2, 3, 5} and R= {(x,y), x + y > 0}, then R is a universal relation.
NOTE: Both Empty and Universal Relations are sometimes called trivial relations.
The relation R in any set A is said to be reflexive; if every element in A is related to itself, i.e., (a, a) ∈ R for every a ∈ A, then a R a is Reflexive.
A relation R in a set A is said to be symmetric if the elements in R can be swapped.
R is symmetric relation if (a, b) ∈ R (b, a) ∈ R, i.e., aRb = bRa a, b, c A.
If S={a, b, c} then relation R6={(a, b), (b, a), (c, c)}
A relation R in a set A is said to be transitive if (a, b) ∈ R and (b, c) ∈ R (a, c) ∈ R, which means that if (a,b) belongs to R and (b,c) belongs to R then (a,c) will also belong to R a, b, c R.
If relation R is reflexive, symmetric and transitive, then it is said to be an equivalence relation.
Notation: a b.
If R={(x, y)/ len (x) = len (y)}
Then R is reflexive: len (a) =len (a)
R is symmetric: len (a)=len (b) and len (b)=len (a)
R is transitive: len (a) = len (b) and len (b) = len (c) then len (a) = len (c)
Therefore, R is an Equivalence Relation.
Equivalence Classes:
Notation: {a} or [a]
Let A be a non-empty set. a ∈ A
An equivalence class is the set of all the points that are in relation to a set A under R.
Functions:
A Function f is a special kind of relation. Let f be the function from set A to set B, i.e., f: A B if every element in A is associated or is mapped with only one element in B.
Set A is called the domain, and Set B is called the Co-domain of the function f.
The range is defined as the set of all possible values of function f.
For example: if x3is a function, all values of x are the domain, and the values of x3 are the range.
Types of Function:
Types of function under class 12 Mathematics chapter 1 notes include
- One-One or Injective function:
Function f is injective if each element of set A is mapped to the distinct element of set B.
Mathematically, f: X → Y is an one-one or injective function, if f(x1) = f(x2) ⇒ x1 = x2 ∀ x1 , x2 ∈ X.
For example: f(x)=3x is a one-one function.
- Onto or Surjective Function:
Function f is said to be Onto if at least one element in the domain is mapped to every element in the codomain. It means for a surjective function, the range, codomain, and the image are equal.
Mathematically, f: X → Y is onto or surjective function if for the given y ∈ Y, ∃ x ∈ X such that f(x) = y.
For example: f(x)=x-2 is a surjective function.
Function f is said to be bijective if it is both injective and surjective.
Mathematically, f: X → Y is one-one and onto or bijective, if f is one-one and onto.
For example: f(x)=3x is a one-one and onto function.
Composition of a Function:
Notation: fog or gof
If Function f: A → B and g: B → C, then the composition of function gof: A → C is given as gof(x) = g(f(x)) for all X in set A
NOTE: fog is used to represent f(g(x))
The function gof is the set of all values of x in the domain of f, where f(x) is in the domain of g.
Properties:
Let f: X → Y, g: Y → Z and h: X → Z, then the functions show the following properties:
- Associativity: f(gh)=(fg)h
- If functions f and g are injective, then gof is also injective
- If functions f and g are surjective, then gof is also surjective
Invertible Function:
A function is said to be invertible if f: X → Y then ∃ g: Y → X such that gof = Ix and fog = Iy.
Here g is known as the inverse of f. It is denoted by f-1.
Condition- If the function f: X → Y is bijective (one-one and onto), then it is said to be invertible.
Therefore, let f: X → Y be a bijective function and f(x) =y, then f-1: X → Y defined as f-1(y)=x, is known as the invertible function of f and is always unique.
Binary Operation
Notation: *
A binary operation is used to associate two elements of a set. They are mathematical operations (addition, subtraction, multiplication and division) performed between two elements of a set.
If S is a non-empty set, let * be the binary operation on S, then a*b is defined for every element belonging to the set S, i.e. for all a,b ∈ S.
We can say that * is a function such that *(x): A × A → A
Example: let there be two real numbers a,b ∈ R then a+b ∈ R.
Properties of Binary Operations:
The properties of binary operations include
1. Commutative property:
The binary operation *: A × A → A is commutative on set A, i.e., for all a, b ∈ A a*b = b*a
2. Associative property
The binary operation *: A × A → A is associative on set A, i.e., for all x, y, z ∈ A x*(y*z) = (x*y)*z
3. Existence of Identity
For the binary operation *: A × A → A, there exists an element e such that a*e = e*a= a for all a, b ∈ A, where e is the identity element of Set A
4. Existence of inverse
For the binary operation *: A × A → A, there exists an element b such that a*b = b*a= a for all a, b ∈ A, where b is the inverse of a denoted by a-1.
Theorems:
- Prove the associativity of the composition of a function. i.e., if f: X → Y, g: Y → Z and h: X → Z, then
show that fo(goh) =(fog)oh
- Prove that two invertible functions f: X → Y, g: Y → Z, then (gof)-1= f-1o g-1
Students may refer to these Class 12 Mathematics Chapter 1 notes while preparing for their examination, as they include all concepts that are a part of the syllabus.
Class 12 Mathematics Chapter 1 Notes: Exercises & Answer Solutions
Chapter 1 Mathematics class 12 notes begin with basic notations and definitions of Relations and Functions. To understand this chapter easily, students need to revise the concepts learnt in Class 11. Students will know about different types of relations, function and their properties, such as associativity, commutativity, the existence of identity and inverse element, etc.
To help students understand this chapter better, Extramarks provides detailed Class 12 Mathematics Chapter 1 notes. It includes all the important definitions, formulae, theorems and properties of this chapter. With the help of the CBSE solutions, students can solve and understand questions using step-by-step and well-explained answers. These Class 12 Mathematics Chapter 1 notes will allow students to retain their learning and reduce simple mistakes.
For quick revision, refer to the Extramarks Exercises & Answer Solutions and MCQ questions under the CBSE sample papers of this chapter from the links below.
Extramarks, an online learning platform, focuses on providing students with a wonderful learning experience. To attain high scores, students are advised to refer to the CBSE revision notes and CBSE previous year question papers and practice as many CBSE extra questions as possible.
NCERT Exemplar Class 12 Mathematics
With the help of the NCERT Exemplar, students can gain an in-depth knowledge and develop a strong foundation in Mathematics. Students can use the Class 12 Mathematics Chapter 1 notes, NCERT Exemplar and other NCERT books to perform well in the exams and pursue their dreams. NCERT Exemplar includes many practice questions of all difficulty levels, which will help in developing analytical and problem-solving abilities. Using the NCERT Exemplar and Class 12 Mathematics Chapter 1 notes, students will learn several tips, tricks and shortcut methods to tackle any question in the examination quickly.
Key Features of Class 12 Mathematics Chapter 1 notes
The key features of Class 12 Mathematics Chapter 1 notes provided by Extramarks are
- These notes are prepared by the subject elites at Extramarks.
- The notes follow the CBSE syllabus and are based on the latest guidelines issued by the board.
- The class 12 Mathematics chapter 1 notes offer the student a wide variety of questions to practice to get acquainted with the concepts in the Class 12 Mathematics Chapter 1.
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