Class 12 Mathematics (041) · Chapter 1 NotesRelations and Functions
Master Class 12 Mathematics Chapter 1: Relations and Functions. Detailed notes on equivalence relations, bijective functions, and function composition.
Chapter 1 of Class 12 Mathematics, Relations and Functions, serves as a bridge between the foundational concepts introduced in Class 11 and the more advanced abstract algebra and calculus studied in higher secondary education. While students are already familiar with the basic notions of domain, co-domain, and range, this chapter delves deeper into the structural properties of relations and functions. It transitions from viewing relations as simple connections between objects to defining them as rigorous mathematical subsets of Cartesian products. Students will explore how specific properties like reflexivity, symmetry, and transitivity combine to form equivalence relations, which are essential for partitioning sets. Furthermore, the chapter categorizes functions based on their mapping characteristics—such as injectivity and surjectivity—and introduces the mechanics of function composition and invertibility. By mastering these concepts, students develop the logical framework necessary to understand complex mathematical transformations and the fundamental nature of operations within various sets.
What you'll learn
1Define and identify empty, universal, and trivial relations in a set.
2Verify the reflexive, symmetric, and transitive properties of a given relation.
3Demonstrate how an equivalence relation partitions a set into disjoint equivalence classes.
4Distinguish between one-one (injective), many-one, and onto (surjective) functions.
5Determine if a function is bijective and explain its significance for invertibility.
6Compute the composition of two functions (gof and fog) and understand its non-commutative nature.
7Establish the conditions under which a function is invertible and find its inverse.
Chapter at a glance
01Relations and Their Types
02Functions and Function Notation
03Binary Operations
04Equivalence Relations
Detailed chapter notes
01
The Nature and Types of Relations
A relation R in a set A is mathematically defined as a subset of the Cartesian product A x A. This abstract definition allows for relations that may not have an obvious linguistic connection. Two extreme cases are the empty relation, where no element is related to any other, and the universal relation, where every element is related to every element in the set. These are often referred to as trivial relations. Most mathematical study focuses on relations that exist between these two extremes, defined by specific algebraic or logical conditions.
Empty RelationR = empty set.
Universal RelationR = A x A.
A relation is a subset of the Cartesian product of sets.
02
Reflexive, Symmetric, and Transitive Properties
To categorize relations effectively, we examine three fundamental properties. A relation is reflexive if every element is related to itself. It is symmetric if the relation between two elements holds regardless of their order. Finally, it is transitive if an element related to a second, and that second related to a third, implies the first is related to the third. Testing these properties is the primary method for determining the structure of a relation within any given set of numbers or objects.
Reflexive(a, a) belongs to R for every a in A.
Symmetric(a1, a2) in R implies (a2, a1) in R.
Transitive(a1, a2) in R and (a2, a3) in R implies (a1, a3) in R.
03
Equivalence Relations and Partitions
An equivalence relation is a powerful tool in mathematics that must simultaneously satisfy the reflexive, symmetric, and transitive properties. When such a relation exists, it naturally divides the set into mutually disjoint subsets known as equivalence classes. Every element in an equivalence class is related to every other element in that same class, but no element is related to an element in a different class. The union of all these disjoint classes reconstructs the original set, effectively partitioning it based on the relation's criteria.
Equivalence RelationMust be reflexive, symmetric, and transitive.
Equivalence Class [a]The set of all elements b related to a.
PartitionsMutually disjoint subsets whose union is the universal set.
04
Injective and Surjective Functions
Functions are specialized relations where every input has exactly one output. An injective (one-one) function ensures that distinct elements in the domain map to distinct elements in the co-domain. Conversely, a many-one function allows different domain elements to share the same image. A surjective (onto) function is one where every element in the co-domain is the image of at least one element from the domain, meaning the range equals the co-domain. These definitions are vital for understanding the mapping strength of a function.
One-one (Injective)f(x1) = f(x2) implies x1 = x2.
Onto (Surjective)For every y in Y, there exists x in X such that f(x) = y.
BijectiveA function that is both one-one and onto.
05
Composition of Functions
Composition is the process of combining two functions such that the output of the first function becomes the input for the second. For functions f: A to B and g: B to C, the composition gof is a function from A to C. It is important to note that function composition is generally not commutative; that is, gof(x) is usually not equal to fog(x). The existence of a composition depends on the range of the first function being a subset of the domain of the second function.
gof(x) is defined as g(f(x)).
The domain of gof is the domain of f.
Composition is a sequential application of mathematical rules.
06
Invertible Functions and Their Properties
A function f: X to Y is invertible if there exists another function g: Y to X that 'undoes' the operation of f. For a function to be invertible, it must be bijective (both one-one and onto). If a function is not one-one, multiple inputs lead to the same output, making it impossible to determine a unique return path. If it is not onto, some outputs have no corresponding input, leaving the inverse undefined for those values. The inverse function is denoted as f^-1.
Invertibility requires the function to be a bijection.
Inverse conditiongof = Identity on X and fog = Identity on Y.
If f is invertible, its inverse is unique.
07
Binary Operations
A binary operation on a set is a calculation that combines two elements of the set (operands) to produce another element of the same set. Formally, it is a function from the Cartesian product A x A to A. While common operations like addition and multiplication are binary operations on the set of real numbers, subtraction and division may not be binary operations on all sets, such as the set of natural numbers, because the result might fall outside the original set.
Binary operationA function mapping A x A to A.
ClosureThe result must belong to the same set.
Common examples include addition and multiplication on real numbers.
Want the complete chapter resources?Topic notes, quizzes and flashcards for Relations and Functions.
Let R be a relation from set A = {1, 2, 3} to set B = {4, 5, 6} defined by R = {(1,4), (2,5), (3,6)}. What is the domain of R?
A{1, 2, 3}
B{4, 5, 6}
C{1, 2, 3, 4, 5, 6}
D{}
Show answer
Answer: (A) {1, 2, 3}
The domain of a relation is the set of all first elements (inputs) in the ordered pairs. Here, the first elements are 1, 2, and 3, so the domain is {1, 2, 3}.
Question 02
If f(x) = 2x + 3, then f(5) equals:
A10
B13
C15
D18
Show answer
Answer: (B) 13
f(5) = 2(5) + 3 = 10 + 3 = 13. Direct substitution of x = 5 into the function.
Question 03
If f(x) = 2x + 3 and g(x) = x - 1, then (f ∘ g)(x) is equal to:
A2x + 1
B2x + 5
C2x - 2
D2x + 3
Show answer
Answer: (A) 2x + 1
(f ∘ g)(x) = f(g(x)) = f(x - 1) = 2(x - 1) + 3 = 2x - 2 + 3 = 2x + 1. Composition means applying g first, then f to the result.
Question 04
Let * be a binary operation on the set of natural numbers defined as a * b = a + b + ab. What is 2 * 3?
A11
B12
C13
D10
Show answer
Answer: (A) 11
Using the definition a * b = a + b + ab, we get 2 * 3 = 2 + 3 + (2)(3) = 2 + 3 + 6 = 11.
Question 05
A relation R on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive. Which of the following relations on the set of all integers is an equivalence relation?
AaRb if a > b
BaRb if a ≡ b (mod 5)
CaRb if a - b = 1
DaRb if a divides b
Show answer
Answer: (B) aRb if a ≡ b (mod 5)
The relation aRb if a ≡ b (mod 5) is reflexive (a ≡ a mod 5), symmetric (if a ≡ b then b ≡ a mod 5), and transitive (if a ≡ b and b ≡ c then a ≡ c mod 5). The other relations fail one or more properties.
Ready for more practice?Unlock the full quiz for this chapter.
What is the difference between a relation and a function?
A relation is any subset of the Cartesian product of two sets, representing any connection between elements. A function is a specific type of relation where every element in the domain is related to exactly one unique element in the co-domain.
How do you prove a relation is an equivalence relation?
To prove a relation is an equivalence relation, you must demonstrate three things: it is reflexive (every element relates to itself), symmetric (if a relates to b, then b relates to a), and transitive (if a relates to b and b relates to c, then a relates to c).
Why must a function be bijective to be invertible?
A function must be one-one so that each output corresponds to exactly one input, allowing a unique reverse mapping. It must be onto so that every element in the co-domain has a pre-image, ensuring the inverse is defined for the entire domain of the inverse function.
What are equivalence classes?
Equivalence classes are disjoint subsets of a set formed by an equivalence relation. All elements within a single class are related to each other under the given relation, and the collection of all such classes covers the entire original set without any overlap.
Is the composition of functions commutative?
Generally, no. Function composition is not commutative, meaning gof is usually not equal to fog. For example, if f(x) = x + 1 and g(x) = x^2, then gof(x) = (x+1)^2 while fog(x) = x^2 + 1.