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Karnataka Class 11 Commerce Maths Relations and Functions

Karnataka Class 11 Commerce Maths Relations and Functions

Karnataka Class 11 Commerce Maths Relations and Functions

Karnataka Class 11 Commerce Maths 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. The word well defined was used by the German Mathematician George Cantor (1845- 1918 A.D) to define a set. He is known as father of set theory. Now-a-days set theory has become basic to most of the concepts in Mathematics.

In our everyday life we come across different types of relations between the objects. The concept of relation has been developed in mathematical form. The word function was introduced by Leibnitz in 1694.

Karnataka Class 11 Commerce Maths Relations and Functions

Function is a special type of relation. Each function is a relation but each relation is not a function. In this lesson we shall discuss some basic definitions and operations involving sets, Cartesian product of two sets, relation between two sets, the conditions under when a relation becomes a function, different types of function and their properties.

Karnataka Class 11 Commerce Maths Relations and Functions syllabus

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 11 Commerce Maths Relations and Functions

Ordered Pairs:

Let a ∈ A, b ∈ B, then the ordered pair of elements a and b is denoted by (a, b) a is the 1st element b is the 2nd element

Equality of ordered pairs:

Two ordered pairs (a, b) and (c, d) are said to be equal of a = c and b = d

Cartesian Product:

Cartesian product of 2 sets A and B is denoted by A B is the set of all the ordered pairs (a, b) ,∀ a ∈ A and b ∈  B.

A × B ={ (a, b), ∀ a ∈ A and b ∈  B}

If A = {a, b}, B = {c, d, e}

A × B = { (a, c), (a, d), (a, e), (b, c), (b, d), (b, e)}

Note:

1. If A = Ø or B = Ø then A × B = Ø

2. If A ≠ Ø or B ≠ Ø then A ≠ B ≠ Ø

3. If A × B = Ø iff A = Ø or B = Ø

4. If A = B then A × B = A2

5. If a ∈ A and b ∈  B. and c ∈  C

Then (A × B × C) = { (a, b, c) ∀  a ∈ A and b ∈  B. and c ∈  C.

Karnataka Class 11 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

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) }

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 11 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 11 Commerce Maths Relations and Functions

Functions:

Set X and Y be two non-empty sets A subset f of X×Y is called as a function if the following conditions hold good.

i) For each x ∈ X , there exists a unique y ∈ Y such that (x, y) ∈ f ii) Elements of x should not be repeated.

Note: 1) If (x1 , y1 ) ∈f and (x2, y2 ) ∈ f then y1 = y2

2) y ∈Y is called as the image of the element x ∈ X

3) x is called as the pre-image of y ∈ Y.

4) The set X is called as the Domain of the function f.

5) The set Y is called as the Range of the function f.

Domain, Co-domain and Range of a function:

Let F : A B i.e. F is a function or mapping from A to B. The set A is called as domain of F and the set B is called as Co-domain of F.

The set consisting of all images is called as the range of F which in symbols is written on F(A)F = { (1, a), (2, b), (3, c), (4, d)}

Domain of F = A= {1, 2, 3, 4} Range of F =F(A)= {a, b, c, d}

Co-domain of F = B= {a, b, c, d)

Note: 1) F (A) ⊆ B

2) Every function is a relation but the converse is not true.

Karnataka Class 11 Commerce Maths Relations and Functions

Different types of Functions:

1. Into functions :

The mapping f : A → B is called as ‘Into’ if there is at least one element of set B which is not an image. (or which has no pre-image in the set A) symbolically we write f : A → B.

2. Many – one function:

The function f : A → B is called as many one function if different elements of set A have the same image in B.

3. One – One function (injective mapping) A function f :

A → B is said to be a one-one function if different elements of sets A have different images in set B.

Symbolically we can write if f (x1 ) = f (x2 ) ⇒ x1  = x2  ∀ x1  , x2 ∈ A.

f (1) = p

f (3) = q

f (5) = r

4. Onto function (surjective mapping) The mapping f : A → B is called as on to if every element of set B is the image of some element of set A.

One ñ one and onto

f (1) = p

f (3) = r

f (2) = q

f (4) = s

5. Bijection Or one – one and onto function

A function f : A → B is said to be bijective of f is both one – one and onto function.

6. one-one into mapping:

A function f = A → B is said to be one ñ one into function if the Range of function is a subset of B

(i) f (x1 ) = f (x2 ) ⇒ x1 = x2 i.e.there must be atleast one element of set B which is not the image of some element of set A.

f (1) = a

f (2) = b

f (3) = c

Element d∈ B is not the image of any element of set A.

7. Many – one onto mapping:

A mapping f : A → B is said to be many-one onto if

i) Range of function is equal to B

(ii) f (x1 ) = f (x2 ) ⇒ x1 ≠ x2 . i.e. 2 or more elements of set A have the same image in set B.

8. Constant function:

If f : A → B and f (x) = k ∀ x ∈ A and k ∈ B i.e. if each and every element of a ∈ A is mapped on to a single element k of the co-domain B then f is called as a constant function.

9. Inverse function :

Let f : A → B be a one-one, onto function from A to B.

Then for each b ∈ B.  f-1(b) ∈ A exists and is unique so that  f-1  : B → A is a function defined by  f-1 (b) = a if f (a) = b

10. Composite functions:

Let A, B and C be any three non-empty lets let f : A → B and g : B → C be any two functions.

Define a function gof : A → C as (gof) a = g(f(a)) ∀ a ∈ A since f(a) ∈ B g(f(a)) ∈ C.

Thus (gof) so obtained is called as composition of f and g.

Similarly we can define the composite function. (fog) x = f (g(x).

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