Relations and Functions Class 12 CBSE Maths Chapter 1

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Class 12 CBSE Maths Syllabus
Chapter 1: Relations and Functions
Chapter 2: Inverse Trigonometric Functions
Chapter 3: Matrices
Chapter 4: Determinants
Chapter 5: Continuity and Differentiability
Chapter 6: Applications of Derivatives
Chapter 7: Integrals
Chapter 8: Applications of Integrals
Chapter 9: Differential Equations
Chapter 10: Vector Algebra
Chapter 11:3D Geometry
Chapter 12: Linear Programming
Chapter 13: Probability

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*Exercise 1.1 Class 12 CBSE Maths Solution PDF

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Chapter 1

Introduction

Class 12 Maths Chapter 1:

A relation from a set A to a set B is a subset of A×BA×B.
A relation on a set A is a subset of A×A. A function which is both one-one and onto function is called a bijective function.
An equivalence relation, if it is reflexive, symmetric and transitive.
The empty relation, if R=ϕ

A relation R is the subset of the cartesian product A×B, where A and B are non-empty sets. The subset of A×B shows the relationship between the elements of A and B. An element of A will relate to an element of B in a particular order. Other elements will also follow that order. Elements of A are the first elements and the elements of B are the second elements. The second element is called the Image of the first element.
Let A be the set of natural numbers and B be the set of whole numbers. x is the first element, and y is the second element of the ordered pair (x, y). Then some of the examples of relations from A to B are –
1. R = {(x, y): x is the next counting number of y, x ∈ A, y ∈ B}
2. R = {(x, y): x is the square of y, x ∈ A, y ∈ B}
3. R = {(x, y): x is the square root of y, x ∈ A, y ∈ B}
4. R = {(x, y): x is multiple of y, x ∈ A, y ∈ B}
5. R = {(x, y): x is the factor of y, x ∈ A, y ∈ B}
In all the above examples, there is a relation between x and y so, we write x R y. We can also write (x, y) ∈ R. All the relations are the subsets of A×B.
Types of Relations
1) Empty Relation – In a set A, if there is a relation R such that the element of A is not related to any element of A then the relation R is called the Empty Relation. It means the subset of A×A is Φ. i.e., R = Φ.
Example – let A be the set of even numbers. Show that the relation R in A given by R = {(x, y): x is the next counting number of y} is the empty relation.
Solution – We know that A is the set of even numbers so, not any number can be the next counting number. Therefore, R is the empty relation, and R = Φ.
2) Universal Relation – In a set A if there is a relation R such that each element of A is related to every element of A then the relation R is called the Universal Relation. It means every subset of A×A is in relation. i.e., R = A×A.
Example – If P is the set of Real numbers. show that the relation R in P given by R = {(a, b): addition of a and b is also a real number} is the universal relation.
Solution – Since P is the set of real numbers so, the addition of any two numbers will be also a real number obviously. Therefore, the relation R is the universal relation and R = P×P.
Note – Sometimes the empty relation and the universal relation are called Trivial Relations.
3) Reflexive Relation – In a set A, if there is a relation R such that every element of A is related to itself then the relation R is said to be the Reflexive Relation. It means for every subset of A×A, (a, a) ∈ R for every a ∈ A.
Example – Let set B = {1, 3, 5, 7} then show that the relation R in B given by R = {(y, y): y is divisible by y} is the reflexive relation.
Solution – We know that every number is divisible by itself so, every number of the set B will also be divisible by itself. Hence, the relation R is the reflexive relation and (y, y) ∈ R for every y ∈ B.
4) Symmetric Relation – In a set A, if there is a relation R such that element a₁ is related to element a₂ and element a₂ is also related to element a₁ then the relation R is known as the Symmetric Relation. For the subset of A×A, if (a₁, a₂) ∈ R implies that (a₂, a₁) ∈ R, for all a₁, a₂ ∈ A.
Example – If set X = {1, 2, 3} then check that the relation R in X given by R = {(1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2)} is the symmetric relation or not.
Solution – Since R = {(1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2)}
We can see that (1, 2) ∈ R and also (2, 1) ∈ R. it means 1 is related to 2 and 2 is also related to 1. All the other elements are also related to each other in symmetric order. Therefore, relation R is the symmetric relation.
5) Transitive Relation – A relation R in a set A is called the Transitive Relation if element a₁ is related to element a₂, element a₂ is related to element a₃, and also element a₁ is related to element a₃. For the subset of A×A, if (a₁, a₂) ∈ R and (a₂, a₃) ∈ R means that (a₁, a₃) ∈ R, for all a₁, a₂, a₃ ∈ A.
Solution – Here, R = {(x, y): x is the brother of y}
In the given relation, if Ram is the brother of Shyam and Shyam is the brother of Swami then Ram is also the brother of Swami. Hence, the relation R is the transitive relation.
6) Equivalence Relation – In a set A, a relation R is said to be an Equivalence Relation if relation R is reflexive, symmetric, and transitive.
Example – If set E is the set of all rational numbers and R is the relation in E given by R = {(a, b): a + b is a rational number}. Show that R is an equivalence relation.
Solution – We know that sum of two rational numbers is always a rational number. Therefore, Relation R is reflexive because the sum of two same rational numbers (a + a) is also a rational number.
Relation R is symmetric because if a + b is a rational number then b + a will be also a rational number.
For three rational numbers a, b, c,
If a + b is a rational number and b + c is a rational number then a + c will be also a rational number. So, relation R is transitive.
Since relation R is reflexive, symmetric, and transitive, therefore, relation R is an equivalence relation.
Example – Let set T = {Ram, Shyam, Swami} then check that the relation R in T given by R = {(x, y): x is the brother of y} is the transitive relation or not.
Relations and functions
Introduction