CBSE Class 12 Mathematics Notes Chapter 1 Relations and Functions

The concept of relation refers to the relationship between two objects or values.

• A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product A x B.
• The set of all the first elements of the ordered pair in a relation R is called the domain of the relation R.
• The set of all the second elements in a relation R is called the range of the relation.

A function is a relationship that specifies that there should only be one output for each input. 

• It is a type of relation (a collection of ordered pairs) that follows a rule, which states that each y-value should be related to just one y-value.
• 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

Class 12 Maths Relations and Functions is of a weightage of 8 marks in the final mathematics examination 2024.

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Class 12 Mathematics Chapter 1 Notes – Relations and Functions

Relation

• Let A and B be two sets.
• Then a relation R from set A to set B is a subset of A x B.
• Thus R is a relation from A to B ⇔ R ⊂ A x B.

Domain

• Let R be a relation from a set A to a set B.
• Then the first set of all the first components or coordinates of the ordered pairs belonging to R is called the domain of R.
• Thus, the domain of R = {a : (a, b) ϵ R}.
• The domain of R ⊂ A.

Range

• Let R be a relation from a set A to a set B.
• Then the set of all second components or coordinates of the ordered pairs belonging to R is called the range of R.
• Thus, the range of R = {b : (a, b) ϵ R}.
• The range of R ⊂ B.

Relation on a Set

• Let A be a non-void set.
• Then a relation from A to itself i.e. a subset of A x A is called a relation on set A.

The Inverse of a Relation

• Let A and B be the two sets, and let R be a relation from a set A to a set B.
• Then, the inverse of R, denoted by R-1 is a relation from B to A and is defined by

R-1 = {(b, a) : (a, b) ϵ R}

• Clearly, (a, b) ϵ R ⇔ (b, a) ϵ R-1
• Also, Domain (R) = Range (R-1) and, Range (R) = Domain (R-1)

Types of Relation

• Different types of Relation are
  • Void Relation
  • Universal Relation
  • Identity Relation
  • Reflexive Relation
  • Symmetric Relation
  • Transitive Relation
  • Antisymmetric Relation

Void Relation

• Let A be a set. Then ɸ ⊂ A x A and so it is a relation on A.
• This relation is called the void or empty relation on set A.
• A relation R on a set A is called a void relation, if no element of A is related to any element of A.

Universal Relation

• Let A be a set. Then A x A ⊂ A x A and so it is a relation on A.
• This relation is called the universal relation on set A.
• A relation R on a set A is called a universal relation, if each element of A is related to every element of A.

Identity Relation

• Let A be a set. Then the relation IA = {(a, a): a ϵ A} on A is called the identity relation on A.
• A relation IA on A is called the identity relation if every element of A is related to itself only.

Reflexive Relation

• A relation R on a set A is said to be reflexive if every element of A is related to itself.
• Thus, R is reflexive ⇔ (a, a) ϵ R for all a ϵ A.
• A relation R on a set A is not reflexive if there exists an element a ϵ A such that (a, a) ∉ R.

Symmetric Relation

• A relation R on a set A is said to be a symmetric relation if

(a, b) ϵ R ⇒ (b, a) ϵ R for all a, b ϵ A

• i.e. aRb ⇒ bRa for all a, b ϵ A

Transitive Relation

• A relation R on a set A is said to be a transitive relation if

(a, b) ϵ R and (b, c) ϵ R ⇒ (a, c) ϵ R for all a, b, c ϵ A

• i.e. aRb and bRc ⇒ aRc for all a, b, c ϵ A

Antisymmetric Relation

A relation R on a set A is said to be an antisymmetric relation if

(a, b) ϵ R and (b, a) ϵ R ⇒ a = b for all a, b ϵ A

Functions

• A function is a relationship that specifies that there should only be one output for each input. 
• It is a type of relation that follows a rule, which states that each y-value should be related to just one y-value.
• Let A and B be two non-empty sets.
• A relation f from A to B i.e. a subset of A x B is called a function from A to B if
  • For each a ϵ A there exists b ϵ B such that (a, b) ϵ f.
  • (a, b) ϵ f and (a, c) ϵ f ⇒ b = c.

Types of Functions

• Different types of functions are
  • One-one Function (Injection)
  • Many-one Function
  • Onto Function (Surjection)
  • One-one Onto Function (Bijection)

One-one Function (Injection)

• A function f: A → B is said to be a one-to-one function or an injection if different elements of A have different images in B.
• Thus, f: A → B is one-one.

One-one Function

Many-one Function

• A function f: A → B is said to be a many-one function if two or more elements of set A have the same images in B.
• Thus, f: A → B is a many-one function if there exists x, y ϵ A such that x ≠ y but f(x) = f(y).

Many-one Function

Onto Function (Surjection)

• A function f: A → B is said to be a onto function or a surjection if every element of set B is the f-image of some element of A i.e. f(A) = B or range of f is the co-domain of f.
• Thus, f: A → B is a surjection if, for each b ϵ B, there exists a ϵ A such that f(a) = b.

Surjection Function

One-one Onto Function (Bijection)

• A function f: A → B is a bijection if it is one-one as well as onto.
• In other words, a function f: A → B is a bijection, if it is
  • One-one i.e. f(x) = f(y) ⇒ x = y for all x, y ϵ A.
  • Onto i.e. for all y ϵ B, there exists x ϵ A such that f(x) = y.

Bijection Function

Composite of Functions

• Let f: A → B and g: B → C be two functions.
• Then the composition of f and g is defined as the function of g o f: A → C given by

g o f(x) = g (f(x)), ∀ x ∈ A

Invertible Function

• A function f: X → Y is defined to be invertible if there exists a function g: Y → X such that g o f = IX and f o g = IY.
• Then the function is called the inverse of f and is denoted by f-1.

Binary Operations

• A binary operation ∗ on a set A is a function ∗: A x A → A.
• We denote ∗ (a, b) by a ∗ b.
• A binary operation ∗ on the set X is called commutative if a ∗ b = b ∗ a for every a, b ϵ X.

There are Some important List Of Top Mathematics Questions On Relations And Functions Asked In CBSE CLASS XII

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