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Karnataka Class 12 Commerce Maths Relations and Functions

Karnataka Class 12 Commerce Maths Relations and Functions

Karnataka Class 12 Commerce Maths Relations and Functions

Karnataka Class 12 Commerce Maths 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

Karnataka Class 12 Commerce Maths Relations and Functions

Relations and Functions – syllabus

  • Cartesian product of sets: Ordered pairs, Cartesian product of sets.
  • Number of elements in the Cartesian product of two finite sets.
  • Cartesian product of the reals with itself (upto R × R × R).
  • Relation: Definition of relation, pictorial diagrams, domain, co-domain and range of a relation and examples
  • Function : Function as a special kind of relation from one set to another.
  • Pictorial representation of a function, domain, co-domain and range of a function.
  • Real valued function of the real variable, domain and range of constant, identity, polynomial rational, modulus, signum and greatest integer functions with their graphs.
  • Algebra of real valued functions:
    Sum, difference, product and quotients of functions with examples.

Karnataka Class 12 Commerce Maths Relations and Functions

Cartesian products of sets

Definition :

Given two non-empty sets A and B, the set of all ordered pairs (x, y), where x ∈ A and y ∈ B is called Cartesian product of A and B;

symbolically, we write A × B = {(x, y) | x ∈ A and y ∈ B} If A = {1, 2, 3} and B = {4, 5},

then A × B = {(1, 4), (2, 4), (3, 4), (1, 5), (2, 5), (3, 5)} and B × A = {(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)}

(i) Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal, i.e. (x, y) = (u, v) if and only if x = u, y = v.

(ii) If n(A) = p and n (B) = q, then n (A × B) = p × q.

(iii) A × A × A = {(a, b, c) : a, b, c ∈ A}.

Here (a, b, c) is called an ordered triplet.

Karnataka Class 12 Commerce Maths Relations and Functions

Relations

A Relation R from a non-empty set A to a non empty set B is a subset of the Cartesian product set A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.

The set of all first elements in a relation R, is called the domain of the relation R, and the set of all second elements called images, is called the range of R.

For example, the set R = {(1, 2), (– 2, 3), ( 1/ 2 , 3)} is a relation; the domain of R = {1, – 2, 1 /2 } and the range of R = {2, 3}.

(i) A relation may be represented either by the Roster form or by the set builder form, or by an arrow diagram which is a visual representation of a relation.

(ii) If n (A) = p, n (B) = q; then the n (A × B) = pq and the total number of possible relations from the set A to set B = 2pq

Karnataka Class 12 Commerce Maths Relations and Functions

Relation :

Let A and B be two empty sets, A relation R from A to B is a subset of A × B i.e. R is a relation from A to B if R ⊆ A × B

1. A relation R from A to A is a subset of A × A

2. If set A has m elements, set B has n elements then A B has mn elements.

3. A × B has 2mn subsets

4. If there exists 2mn relations from A to B Then there are 2mn relations from B to A

5. If R is a relation from A to B and (x, y) ∈ R then this is denoted by x Ry

Karnataka Class 12 Commerce Maths Relations and Functions

Domain and Range of a relation:

Let A and B be 2 non-empty sets and R be a relation from A to B i.e. R ⊆ A × B.

The domain of R is defined as the collection of all the first elements of the ordered pairs (a, b) ∈ R.

i.e. Domain of R = {a ∈ A :(a, b) ∈ R}

The Range of R is defined as the set of all the second elements of the ordered pairs (a, b) ∈ R

i.e. Range of R = {b ∈ R : (a, b) ∈ R }

Example: Let A = {b, c, d} B = {c, d, e} If R is a relation from A to B defined by is defined as “next letter in English Alphabet”

Then R-1 = { (b, c), (c, d), (d, e) }

Karnataka Class 12 Commerce Maths Relations and Functions

Inverse relation:

Let R be a relation from A to B. The inverse relation of R is denoted by R-1 and is a relation from B to A

i.e. sub set of (B × A) defined as follows

R-1 = { ( y, x ) ∀  (x, y) ∈ R } Let A = {1, 2} B = {a, b} If R is a relation from A to B defined by R = { (1, a), (1, b), (2, a)}

Then R-1 = { (a, 1), (b, 1), (a, 2) }.

Karnataka Class 12 Commerce Maths Relations and Functions

Types of relations:

1) Identity relation: 

Let A be a non empty set the relation IA  defined by IA = { (a, a) : a ∈ A} is called the identity relation on A.

2) Null relation (void relation):

Let A be a non-empty set. We know that Ø ⊂ A × A and hence Ø is a relation on A. This relation is called as the null relation on A.

3) Universal Relation:

Let A be a non empty set . we know that (A × A) ⊆ (A × A) and hence AA is a relation on A. This relation is called as Universal Relation on A.

4) Reflexive Relation:

A relation R on a non empty set A is called a reflexive relation if (a, a) ∈ R ∀ a ∈  R

If A = { a, b, c } R1 = {(a, a), (b, b)} R2 = { (a, a), (b, b), (c. c) } are reflexive relations on A

Note : The identity relation and universal relation on a non empty set are reflexive relations.

5) Symmetric Relation:

A relation R on a non empty set A is called a symmetric relation, if (a, b) ∈ R ⇒ (b, a) ∈ R

Example: A = {1, 2, 3, 4, 5 } R1 = { (2, 3), (3, 2), (3, 4), (4, 3) } And R2 = {(1, 5), (5, 1), (2, 5), (5, 2) } are symmetric relations on A.

Note : The universal relation on a non empty set is a symmetric relation.

6) Transitive Relation:

A relation R on a non empty set A is called as a Transitive Relation if (a, b), (b, c) ∈ R ⇒ (a, c) ∈ R

Example: Let R be a relation on the set of naturals defined by ‘is a factor of’

xRy ⇒ x is a factor of y

yRz ⇒y is a factor of z

 ⇒ xRz ⇒ x is a factor of z

7) Equivalence Relation:

A relation R on a non empty set A is called an equivalence relation if it is reflexive, symmetric and transitive.

On set ‘L’ of straight lines in a plane

i) Reflexive Relation: line ‘l1‘ is parallel to itself; ∴ R is reflexive.

ii) Symmetric relation: l1 is parallel to line l2 then l2 is parallel to line l1 (l1 , l2 ) ∈ R ⇒ (l2 , l1) ∈ R;  ∴ R is symmetric relation.

iii) Transitive Relation: If a line l1 is parallel to l2 and a line l2 is parallel to l3¸ then we know l1 is parallel to l3 .

If (l1 , l2 ), (l2 , l3 )∈ R ⇒ (l1 , l3 )∈ R

R is transitive. Since ‘R’ is reflexive, symmetric and transitive,

∴ it is an equivalence relation.

8) Anti Symmetric Relation:

A relation R on a non empty set A is called as anti symmetric relation if (a, b), (b, a) ∈ R ⇒ a = b

Example: Consider the relation R defined by ‘is less than or equal to on the set of integers,

If x, y ∈ R such that x ≤ y and y ≤ x then x = y.

∴ R is an anti symmetric relation.

Karnataka Class 12 Commerce Maths Relations and Functions

Functions

A relation f from a set A to a set B is said to be function if every element of set A has one and only one image in set B. In other words, a function f is a relation such that no two pairs in the relation has the same first element. The notation f : X → Y means that f is a function from X to Y. X is called the domain of f and Y is called the co-domain of f.

Given an element x ∈ X, there is a unique element y in Y that is related to x. The unique element y to which f relates x is denoted by f (x) and is called f of x, or the value of f at x, or the image of x under f.

The set of all values of f (x) taken together is called the range of f or image of X under f. Symbolically. range of f = { y ∈ Y | y = f (x), for some x in X}

Karnataka Class 12 Commerce Maths Relations and Functions

Definition :

A function which has either R or one of its subsets as its range, is called a real valued function. Further, if its domain is also either R or a subset of R, it is called a real function.

Karnataka Class 12 Commerce Maths Relations and Functions

Some specific types of functions

(i) Identity function:

The function f : R → R defined by y = f (x) = x for each x ∈ R is called the identity function. Domain of f = R Range of f = R

(ii) Constant function:

The function f : R → R defined by y = f (x) = C, x ∈ R, where C is a constant ∈ R, is a constant function. Domain of f = R Range of f = {C}

(iii) Polynomial function:

A real valued function f : R → R defined by y = f (x) = a0 + a1 x + …+ an xn , where n ∈ N, and a0 , a1, a2 …an ∈ R, for each x ∈ R, is called Polynomial functions.

(iv) Rational function:

These are the real functions of the type f(x)/g(x) , where f (x) and g (x) are polynomial functions of x defined in a domain, where g(x) ≠ 0.

For example f : R – {– 2} → R defined by f (x) = (x + 1)/(x + 2) , ∀ x ∈ R – {– 2 }is a rational function.

(v) The Modulus function:

The real function f : R → R defined by f (x) = ΙxΙ = {x, x≥0 and -x, x<0, ∀x ∈ R is called the modulus function.

Domain of f = R

Range of f = R+ ∪ {0}

(vi) Signum function:

The real function f : R → R defined by f(x) = {|x| ,  x ≠ 0 and 0 ,x = 0} = 1(x>0), 0(x=0), – 1(x<0), is called the signum function.

Domain of f = R,

Range of f = {1, 0, – 1}

(vii) Greatest integer function:

The real function f : R → R defined by f (x) = [x], x ∈R assumes the value of the greatest integer less than or equal to x, is called the greatest integer function. Thus

f (x) = [x] = – 1 for – 1 ≤ x < 0

f (x) = [x] = 0 for 0 ≤ x < 1

f (x) = [x] = 1 for 1 ≤ x < 2

f (x) = [x] = 2 for 2 ≤ x < 3 and so on.

Karnataka Class 12 Commerce Maths Relations and Functions

Algebra of real functions

(i) Addition of two real functions

Let f : X → R and g : X → R be any two real functions, where X ∈ R. Then we define ( f + g) : X → R by ( f + g) (x) = f (x) + g (x), for all x ∈ X.

(ii) Subtraction of a real function from another

Let f : X → R and g : X → R be any two real functions, where X ⊆ R. Then, we define (f – g) : X → R by (f – g) (x) = f (x) – g (x), for all x ∈ X.

(iii) Multiplication by a Scalar

Let f : X → R be a real function and α be any scalar belonging to R. Then the product αf is function from X to R defined by (α f ) (x) = α f (x), x ∈ X.

(iv) Multiplication of two real functions

Let f : X → R and g : x → R be any two real functions, where X ⊆ R. Then product of these two functions i.e. f g : X → R is defined by ( f g ) (x) = f (x) g (x) ∀x ∈ X.

(v) Quotient of two real function

Let f and g be two real functions defined from X → R. The quotient of f by g denoted by f g is a function defined from X → R as (f/g) (x) = f(x) /g(x)  provided g (x) ≠ 0, x ∈ X.

Karnataka Class 12 Commerce Maths Relations and Functions

Note

Domain of sum function f + g, difference function f – g and product function

fg = {x : x ∈D f ∩ Dg }

where Df = Domain of function f

Dg = Domain of function g

Domain of quotient function f / g = {x : x ∈D f ∩ Dg and g (x) ≠ 0}.

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