FOR INDIA'S BEST CA CS CMA VIDEO CLASSES OR Take This Quiz & Predict Your Score in the coming CA CS or CMA Exam!
• How important it is for you to pass the exam in this attempt?
• What percentage of course you have finished well so far roughly?
• How many hours you study in a day?
• How many times you have revised the topics you have finished
• Have you taken online or pen drive or live class from a renowned faculty?
• What percentage of the classes you have watched?
• Have you attempted mock tests or practice tests yet?
• Are you planning to attempt mock tests conducted by external bodies- ICAI, ICSI, ICMAI or other institute?
• How many tests you have taken?
• Did you manage to finish the test papers on time?
• Are you strictly following study material provided by the exam conducting authority such as ICAI/ICSI/ICMAI/Other Body?
• How is your health in general?
• How is your food habit?
• Any interest in yoga or exercise or play sports regularly?
• Planning to sleep well nights before the exams?
• Planning to have light food and water before exams?

# Karnataka Class 12 Commerce Maths Unit I Relations And Functions

## Karnataka Class 12 Commerce Maths Unit I Relations And Functions

### Karnataka Class 12 Commerce Maths Unit I Relations And Functions

Karnataka Class 12 Commerce Maths Unit I Relations And Functions : A set is a collection of well defined objects. For a collection to be a set it is necessary that it should be well defined. Function is a special type of relation. Each function is a relation but each relation is not a function.

### Karnataka Class 12 Commerce Maths Unit I Relations And Functions

RELATIONS

Consider the following example : A={Mohan, Sohan, David, Karim} B={Rita, Marry, Fatima} Suppose Rita has two brothers Mohan and Sohan, Marry has one brother David, and Fatima has one brother Karim.

If we define a relation R ” is a brother of” between the elements of A and B then clearly. Mohan R Rita, Sohan R Rita, David R Marry, Karim R Fatima.

After omiting R between two names these can be written in the form of ordered pairs as : (Mohan, Rita), (Sohan, Rita), (David, Marry), (Karima, Fatima).

The above information can also be written in the form of a set R of ordered pairs as R= {(Mohan, Rita), (Sohan, Rita), (David, Marry), Karim, Fatima}

Clearly R Í A´B, i.e.R = {(a,b):aÎ Î A,b B and aRb} If A and B are two sets then a relation R from A to B is a sub set of A×B.

If (i) R = f , R is called a void relation.

(ii) R=A×B, R is called a universal relation.

(iii) If R is a relation defined from A to A, it is called a relation defined on A.

(iv) R = { (a,a)aA ” Î } , is called the identity relation

### Karnataka Class 12 Commerce Maths Unit I Relations And Functions

DEFINITION OF A FUNCTION

Consider the relation f : {(a, 1), (b, 2), (c, 3), (d, 5)} In this relation we see that each element of A has a unique image in B This relation f from set A to B where every element of A has a unique image in B is defined as a function from A to B. So we observe that in a function no two ordered pairs have the same first element. We also see that \$ an element Î B, i.e., 4 which does not have its pre image in A. Thus here:

(i) the set B will be termed as co-domain and

(ii) the set {1, 2, 3, 5} is called the range. From the above we can conclude that range is a subset of co-domain.

Symbolically, this function can be written as f : A ® B or A f ¾¾¾® B

### Karnataka Class 12 Commerce Maths Unit I Relations And Functions

Relations and Functions

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 Unit I 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 Unit I 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}.

RELATIONS AND FUNCTIONS

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 Unit I 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}

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 Unit I 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) = [x] = , {x, x>/= 0;- 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, x not equal to 0; 0, x= o;} = 1, if x>0; = 0, if x=0; = -1, if 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

[x] = 1 for 1 ≤ x < 2

[x] = 2 for 2 ≤ x < 3 and so on

### Karnataka Class 12 Commerce Maths Unit I 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.

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}

Example of how to answer a question paper can be seen here : Steps to answer question paper

How to Allot CBSE Class 12 Commerce Marks : CBSE Class 12 Commerce Allotment of Marks

To download CBSE Class 12 Commerce NCERT Books syllabus check here

CBSE Class 12 Commerce syllabus

CBSE Class 12 Commerce syllabus (2)

For more on CBSE Class 12 Commerce log onto www.cakart.in