Relations and Functions
NCERT Class 12 Mathematics - Chapter 1: Relations and Functions
Topics Covered
- Types of Relations
- Types of Functions
- Composition of Functions
- Invertible Functions
- Binary Operations
Types of Relations
Relations are used to define the relationship between two sets. There are various types of relations in mathematics:
- Reflexive Relation
- Symmetric Relation
- Transitive Relation
- Equivalence Relation
Example
If R is a relation on set A = {1, 2, 3} defined by R = {(1, 1), (2, 2), (3, 3)}, then R is reflexive.
Word Meaning: Reflexive Relation
A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A.
Types of Functions
Functions are special types of relations that associate each element in one set with exactly one element in another set. Types of functions include:
- One-to-One Function (Injective)
- Onto Function (Surjective)
- One-to-One Onto Function (Bijective)
Example
If f: A → B is defined by f(x) = 2x for A = {1, 2, 3} and B = {2, 4, 6}, then f is one-to-one and onto.
Word Meaning: Bijective Function
A function f: A → B is called bijective if it is both injective (one-to-one) and surjective (onto).
Composition of Functions
The composition of functions is a way to combine two functions. If f: A → B and g: B → C, then the composition g ∘ f: A → C is defined by (g ∘ f)(x) = g(f(x)).
Example
If f(x) = x2 and g(x) = x + 1, then (g ∘ f)(x) = g(f(x)) = (x2) + 1.
Word Meaning: Composition of Functions
The operation of combining two functions such that the output of one function becomes the input of the other.
Invertible Functions
A function f: A → B is invertible if there exists a function g: B → A such that g(f(x)) = x for all x in A and f(g(y)) = y for all y in B. The function g is called the inverse of f, denoted by f-1.
Example
If f(x) = 2x and g(x) = x/2, then f and g are inverses of each other, i.e., f-1(x) = x/2.
Word Meaning: Invertible Function
A function that has an inverse, such that applying the function and its inverse returns the original value.
Binary Operations
A binary operation * on a set A is a function *: A × A → A. Common examples of binary operations are addition, subtraction, multiplication, and division.
Example
Addition (+) is a binary operation on the set of real numbers, as for any two real numbers a and b, a + b is also a real number.
Word Meaning: Binary Operation
An operation that combines two elements (operands) to produce another element.
FAQs
1. What is a reflexive relation?
A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A.
2. What is a symmetric relation?
A relation R on a set A is called symmetric if (a, b) ∈ R implies (b, a) ∈ R for all a, b ∈ A.
3. What is a transitive relation?
A relation R on a set A is called transitive if (a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R for all a, b, c ∈ A.
4. What is an equivalence relation?
A relation that is reflexive, symmetric, and transitive is called an equivalence relation.
5. What is an injective function?
An injective function is a one-to-one function where each element of the domain maps to a unique element of the codomain.
6. What is a surjective function?
A surjective function is an onto function where each element of the codomain has at least one element of the domain mapping to it.
7. What is a bijective function?
A bijective function is both injective and surjective, meaning it is a one-to-one onto function.
8. How do you compose two functions?
The composition of two functions f and g is denoted by g ∘ f and defined as (g ∘ f)(x) = g(f(x)).
9. What is an invertible function?
A function f: A → B is invertible if there exists a function g: B → A such that g(f(x)) = x for all x in A and f(g(y)) = y for all y in B.
10. What is a binary operation?
A binary operation on a set A is a function that combines two elements from A to produce another element from A.
11. What are the common types of relations?
The common types of relations are reflexive, symmetric, transitive, and equivalence relations.
12. What is the significance of a bijective function?
A bijective function ensures a perfect pairing between the domain and codomain, making it invertible.
13. How do you determine if a function is invertible?
A function is invertible if it is bijective, meaning it is both injective and surjective.
14. What is the role of composition in functions?
Composition allows the combination of two functions to form a new function, providing a way to build complex functions from simpler ones.
15. Can a function be both injective and surjective?
Yes, such a function is called bijective.
16. What is an example of a reflexive relation?
A relation R on set A = {1, 2, 3} defined by R = {(1, 1), (2, 2), (3, 3)} is reflexive.
17. What is an example of a binary operation?
Addition is a binary operation on the set of real numbers.
18. What is an example of an injective function?
If f(x) = 2x, then f is injective as each input maps to a unique output.
19. What is an example of a surjective function?
If f(x) = x^2 and the codomain is non-negative real numbers, then f is surjective.
20. What is an example of an invertible function?
If f(x) = 2x and g(x) = x/2, then f and g are inverses of each other.