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CBSE Class 12 Commerce Mathematics Unit I Relations and Functions

CBSE Class 12 Commerce Mathematics Unit I Relations and Functions

CBSE Class 12 Commerce Mathematics Unit I Relations and Functions

CBSE Class 12 Commerce Mathematics Unit I Relations and Functions:- Recall that the notion of relations and functions, domain, co-domain and range have been introduced in Class XI along with different types of specific real valued functions and their graphs. The concept of the term ‘relation’ in mathematics has been drawn from the meaning of relation in English language, according to which two objects or quantities are related if there is a recognisable connection or link between the two objects or quantities. Let A be the set of students of Class XII of a school and B be the set of students of Class XI of the same school.

CBSE Class 12 Commerce Mathematics Unit I Relations and Functions

Then some of the examples of relations from A to B are
(i) {(a, b) ∈ A × B: a is brother of b},
(ii) {(a, b) ∈ A × B: a is sister of b},
(iii) {(a, b) ∈ A × B: age of a is greater than age of b},
(iv) {(a, b) ∈ A × B: total marks obtained by a in the final examination is less than the total marks obtained by b in the final examination},
(v) {(a, b) ∈ A × B: a lives in the same locality as b}. However, abstracting from this, we define mathematically a relation R from A to B as an arbitrary subset of A × B.
If (a, b) ∈ R, we say that a is related to b under the relation R and we write as a R b. In general, (a, b) ∈ R, we do not bother whether there is a recognisable connection or link between a and b. As seen in Class XI, functions are special kind of relations.

CBSE Class 12 Commerce Mathematics Unit I Relations and Functions

we would like to study different types of relations. We know that a relation in a set A is a subset of A × A. Thus, the empty set φ and A × A are two
extreme relations.

For illustration, consider a relation R in the set A = {1, 2, 3, 4} given by R = {(a, b): a – b = 10}. This is the empty set, as no pair (a, b) satisfies the condition a – b = 10.

Similarly, R′ = {(a, b) : | a – b | ≥ 0} is the whole set A × A, as all pairs(a, b) in A × A satisfy | a – b | ≥ 0. These two extreme examples lead us to the
following definitions.
Definition 1 A relation R in a set A is called empty relation, if no element of A is related to any element of A, i.e., R = φ ⊂ A × A.
Definition 2 A relation R in a set A is called universal relation, if each element of A is related to every element of A, i.e., R = A × A. Both the empty relation and the universal relation are some times called trivial relations.

Definition 3 A relation R in a set A is called
(i) reflexive, if (a, a) ∈ R, for every a∈ A,
(ii) symmetric, if (a1, a2) ∈ R implies that (a2, a1)∈ R, for all a1, a2 ∈ A.
(iii) transitive, if (a1, a2) ∈ R and (a2, a3)∈ R implies that (a1, a3)∈ R, for all a1, a2, a3 ∈ A.

Definition 4 A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.

CBSE Class 12 Commerce Mathematics Unit I Relations and Functions:-Types of Functions

The notion of a function along with some special functions like identity function, constant function, polynomial function, rational function, modulus function, signum function etc. along with their graphs have been given in Class XI. Addition, subtraction, multiplication and division of two functions have also been studied. As the concept of function is of paramount importance in mathematics and among other disciplines as well, we would like to extend our study about function from where we finished earlier. In this section, we would like to study different types of functions. Consider the functions f1, f2, f3 and f4 given by the following diagrams.
In Fig 1.2, we observe that the images of distinct elements of X1 under the function f1 are distinct, but the image of two distinct elements 1 and 2 of X1 under f2 is same, namely b. Further, there are some elements like e and f in X2 which are not images of any element of X1 under f1, while all elements of X3 are images of some elements of X1 under f3.

CBSE Class 12 Commerce Mathematics Unit I Relations and Functions

The above observations lead to the following definitions:

Definition 5 A function f : X → Y is defined to be one-one (or injective), if the images of distinct elements of X under f are distinct, i.e., for every x1, x2 ∈ X, f (x1) = f (x2) implies x1 = x2. Otherwise, f is called many-one. The function f1 and f4 in Fig 1.2 (i) and (iv) are one-one and the function f2 and f3 in Fig 1.2 (ii) and (iii) are many-one.
Definition 6 A function f : X → Y is said to be onto (or surjective), if every element of Y is the image of some element of X under f, i.e., for every y ∈ Y, there exists an element x in X such that f (x) = y. The function f3 and f4 in Fig 1.2 (iii), (iv) are onto and the function f1 in Fig 1.2 (i) is not onto as elements e, f in X2 are not the image of any element in X1 under f1.

Definition 7 A function f : X → Y is said to be one-one and onto (or bijective), if f is both one-one and onto.

CBSE Class 12 Commerce Mathematics Unit I Relations and Functions:- Composition of Functions and Invertible Function

In this section, we will study composition of functions and the inverse of a bijective function. Consider the set A of all students, who appeared in Class X of a Board Examination in 2006. Each student appearing in the Board Examination is assigned a roll number by the Board which is written by the students in the answer script at the time of examination. In order to have confidentiality, the Board arranges to deface the roll numbers of students in the answer scripts and assigns a fake code number to each roll number. Let B ⊂ N be the set of all roll numbers and C ⊂ N be the set of all code numbers. This gives rise to two functions f : A → B and g : B → C given by f (a) = theroll number assigned to the student a and g(b) = the code number assigned to the roll number b. In this process each student is assigned a roll number through the function f and each roll number is assigned a code number through the function g.

CBSE Class 12 Commerce Mathematics Unit I Relations and Functions

Thus, by the combination of these two functions, each student is eventually attached a code number. This leads to the following definition:

Definition 8 Let f : A → B and g : B → C be two functions. Then the composition of f and g, denoted by gof, is defined as the function gof : A → C given by gof (x) = g(f (x)), ∀ x ∈ A.

Definition 9 A function f : X → Y is defined to be invertible, if there exists a function g : Y → X such that gof = IX and fog = IY. The function g is called the inverse of f and is denoted by f –1.
Thus, if f is invertible, then f must be one-one and onto and conversely, if f is one-one and onto, then f must be invertible. This fact significantly helps for proving a function f to be invertible by showing that f is one-one and onto, specially when the actual inverse of f is not to be determined.

CBSE Class 12 Commerce Mathematics Unit I Relations and Functions:- Binary Operations

Right from the school days, you must have come across four fundamental operations namely addition, subtraction, multiplication and division. The main feature of these operations is that given any two numbers a and b, we associate another number a + b or a – b or ab or a b , b ≠ 0. It is to be noted that only two numbers can be added or multiplied at a time. When we need to add three numbers, we first add two numbers and the result is then added to the third number. Thus, addition, multiplication, subtraction and division are examples of binary operation, as ‘binary’ means two. If we want to have a general definition which can cover all these four operations, then the set of numbers is to be replaced by an arbitrary set X and then general binary operation is nothing but association of any pair of elements a, b from X to another element of X.

CBSE Class 12 Commerce Mathematics Unit I Relations and Functions

This gives rise to a general definition as follows:

Definition 10 A binary operation ∗ on a set A is a function ∗ : A × A → A. We denote ∗ (a, b) by a ∗ b.

Definition 11 A binary operation ∗ on the set X is called commutative, if a ∗ b = b ∗ a, for every a, b ∈ X.

Definition 12 A binary operation ∗ : A × A → A is said to be associative if (a ∗ b) ∗ c = a ∗ (b ∗ c), ∀ a, b, c, ∈ A.

Definition 13 Given a binary operation ∗ : A × A → A, an element e ∈ A, if it exists,
is called identity for the operation ∗, if a ∗ e = a = e ∗ a, ∀ a ∈ A.

Definition 14 Given a binary operation ∗ : A × A → A with the identity element e in A, an element a ∈ A is said to be invertible with respect to the operation ∗, if there exists an element b in A such that a ∗ b = e = b ∗ a and b is called the inverse of a and is denoted by a–1.

CBSE Class 12 Commerce Mathematics Unit I Relations and Functions

Different types of relations and equivalence relation, composition of functions, invertible functions and binary operations. The main features of this chapter are as follows:

  • Empty relation is the relation R in X given by R = φ ⊂ X × X.
  • Universal relation is the relation R in X given by R = X × X.
  • Reflexive relation R in X is a relation with (a, a) ∈ R ∀ a ∈ X.
  • Symmetric relation R in X is a relation satisfying (a, b) ∈ R implies (b, a) ∈ R.
  • Transitive relation R in X is a relation satisfying (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R.
  • Equivalence relation R in X is a relation which is reflexive, symmetric and transitive.
  • Equivalence class [a] containing a ∈ X for an equivalence relation R in X is the subset of X containing all elements b related to a.
  • A function f : X → Y is one-one (or injective) if f (x1) = f (x2) ⇒ x1 = x2 ∀ x1, x2 ∈ X.
  • A function f : X → Y is onto (or surjective) if given any y ∈ Y, ∃ x ∈ X such that f (x) = y.
  • A function f : X → Y is one-one and onto (or bijective), if f is both one-one and onto.
  • The composition of functions f : A → B and g : B → C is the function gof : A → C given by gof (x) = g(f (x)) ∀ x ∈ A.
  • A function f : X → Y is invertible if ∃ g : Y → X such that gof = IX and fog = IY.
  • A function f : X → Y is invertible if and only if f is one-one and onto.
  • Given a finite set X, a function f : X → X is one-one (respectively onto) if and only if f is onto (respectively one-one). This is the characteristic property of a finite set. This is not true for infinite set
  • A binary operation ∗ on a set A is a function ∗ from A × A to A.
  • An element e ∈ X is the identity element for binary operation ∗ : X × X → X, if a ∗ e = a = e ∗ a ∀ a ∈ X.
  • An element a ∈ X is invertible for binary operation ∗ : X × X → X, if there exists b ∈ X such that a ∗ b = e = b ∗ a where, e is the identity for the binary operation ∗. The element b is called inverse of a and is denoted by a–1.
  • An operation ∗ on X is commutative if a ∗ b = b ∗ a ∀ a, b in X.
  • An operation ∗ on X is associative if (a ∗ b) ∗ c = a ∗ (b ∗ c)∀ a, b, c in X.

CBSE Class 12 Commerce Mathematics Unit I Relations and Functions

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