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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}

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