CONTENTCONTENT

• 1.ORDERED PAIRS

• 2.CARTESIAN PRODUCT OF SETS

• 3.RELATIONS

• 4.FUNCTIONS

• 5.ILLUSTRATIONS

• 6.REAL FUNCTIONS AND THEIR GRAPHS

Ordered PairOrdered Pair

• A pair of objects listed in a specific order is called ordered pair.

• It is written by listing the two objects in the specified order, separating by a comma and enclosing the pair in parentheses.

• Eg: (5,7) is an ordered pair with 5 as the first element and 7 as the second element.

• Two ordered pair are said to be equal if their corresponding elements are equal. i.e., (a,b) = (c,d) if a = c and b = d

• The sets {a,b} and {b,a} are equal but the ordered pairs (a,b) and (b,a) are not equal.

Cartesian Product Of SetsCartesian Product Of Sets

• The Cartesian product of two non empty sets A and B is defined as the set of all ordered pairs (a,b), where a є A, b є B. The Cartesian product of sets A and B is denoted by A x B. Thus AxB = {(a,b) : a є A and b є B}

• If A = Ф or B = Ф, then we define A x B = Ф

• Eg: If A = {2,4,6} and B = {1,2} then

A x B = {(2,1), (2,2), (4,1), (4,2), (6,1), (6,2)}

B x A = {(1,2), (1,4), (1,6), (2,2), (2,4), (2,6)}

• No of elements in the Cartesian product of two finite sets A and B is given by n(A x B) = n(A).n(B) in the above example n(A)=3 and n(B)=2 n(A x B) = 3 * 2 = 6

RelationsRelations• Let P = {a,b,c} and Q = {Ali, Bhanu, Binoy, Chandra, Divya}. P x Q

contains 15 ordered pairs given by P x Q = {(a, Ali), (a, Bhanu), (a, Binoy), ….. (c,Divya)}.

• We can now obtain a subset of P x Q by introducing a relation R between the first element x and the second element y of each ordered pair (x,y) as R = {(x,y): x is the first letter of the name y, x є P, y є Q}. Then R = {(a, Ali), (b, Bhanu), (b, Binoy), (c, Chandra)}

• A relation R from a non-empty set A to anon-empty set B is a subset of the cartesian product A x B.

• The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R.

• The set of all second elements in a relation R from a set A to a set B is called the range of the relation R. The whole set B is called the Codomain of the relation R. Range codomain

Number of RelationsNumber of Relations• Let A and B be two non-empty finite sets consisting of m and n

elements respectively.

A x B contain mn ordered pairs.

Total number of subsets of A x B is 2mn. Since each relation from A x B is a subset of A x B, the total number of relations from A to B is 2mn

• Eg: Let A = {1,2,3,4,5,6,7,8} and R = {(x,2x + 1): x є A}

• When x = 1, 2x + 1 = 3 є A (1,3) є R

When x = 2, 2x + 1 = 5 є A (2,5) є R

When x = 4, 2x + 1 = 9 A (4,9) R

Similarly (5,11) R, (6,13) R and (7,15) R

R = {(1,3), (2,5), (3,7)}

R 1 2 3 4 5 6 7 8

1 0 0 1 0 0 0 0 0

2 0 0 0 0 1 0 0 0

3 0 0 0 0 0 0 1 0

4 0 0 0 0 0 0 0 0

5 0 0 0 0 0 0 0 0

6 0 0 0 0 0 0 0 0

7 0 0 0 0 0 0 0 0

8 0 0 0 0 0 0 0 0

1. Tabular Diagram for R

A

A

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

A AR

2. Arrow Diagram

FunctionsFunctions• A relation F from the set A to a set B is said to be a function if

– Every element of set A has one and only one image in set B

– A function f is a relation from a non-empty set A to a non-empty set B such that 1) The domain of f is A. 2) No two distinct ordered pairs in f have the same first element.

Eg: Let f assign to each country in the world its capital city, since each country in the world has a capital and exactly one capital, f is a function

f (India) = Delhi, f (England) = London,

If f is as function from A to B, then we write f : A B

If the element x of A corresponds to y(єB) under the function f, then we say

that y is the image of x under f and we write f (x) = y. We also say that x is a

pre-image of y.

x y = f(x)

A B

f

• Eg: Let A = {1,2,3,4} and B = {1,6,8,11,15}. Which of the following are functions from A to B?

1. f : A B defined by f(1) = 1, f(2) = 6, f(3) = 8, f(4) = 8.

2. f : A B defined by f(1) = 1, f(2) = 6, f(3) = 15.

3. f : A B defined by f(1) = 6, f(2) = 6, f(3) = 6, f(4) = 6.

4. f : A B defined by f(1) = 1, f(2) = 6, f(2) = 8, f(3) = 8. f(4) = 11.

5. f : A B defined by f(1) = 1, f(2) = 8, f(3) = 11, f(4) = 15.

Pictorial Representation of a FunctionPictorial Representation of a Function

• Let A = {1,2,3,4} and B = {x,y,z}. Let f : A B be a function defined on f(1) = x, f(2) = y, f(3) = y, f(4) = x. This function is represented by using an arrow diagram.

1234

X

y

z

A Bf

Illustration 1Illustration 1

• Let A = {2, 3, 4} B = {1, 3, 6, 8}. f is defined such that f(2) = 3, f(3) = 8, f(4) = 1. Here f is a function

• Domain of f = A = {2, 3, 4}

• Co-domain of f = B = {1, 3, 6, 8}

• Range of f = {3, 8, 1}

• Range f co-domain of f

2

3

4

1

3

6

8

BA f

Illustration 2Illustration 2

• Let X = {3, 6, 8} Y = {a, b, c}.

• f : X Y defined by • f(3) = a, f(6) = c.• Here f is not a function

because there is no element of Y which correspond to 8 of X

3

6

8

a

b

c

YX f

Illustration 3Illustration 3

• Let X = {1, 5, 7} Y = {2, 3, 4, 7}.

• f : X Y defined by • f(1) = 4, f(5) = 4.• f(7) = 3, f(7) = 7.• Here f is not a function

because for 7 of X, there are two images in Y

1

5

7

2

3

4

7

FX f

Illustration 4Illustration 4

• Let X = {2,3,4,7} Y = {1,2,3,4,5,6,7}.

• f : X Y defined by

• f(2) = 5, f(3) = 3.

• f(4) = 3, f(7) = 6.

• Here f is a function because to each element of X there correspond exactly one element of Y.

• Note: Here the elements 3 and 4 of X are corresponding to the same element 3 of Y. This situation is not violating the definition of a function.

2

3

4

7

1234567

YX g

Real Valued FunctionReal Valued Function

• Let f be a function from the set A to the set B. If A and B are sub sets of real number system R then f is called a real valued function of a real variable. In short we call such a situation as a real function.

• Eg: f : R R defined by f(x) = x2 + 3x + 7, x є R is a real function.

Some Real Functions and their GraphsSome Real Functions and their Graphs

1. Constant function Def: A function f : R R is

called a constant function if there exists an element k є R such that f(x) = k x є R

Rule: f(x) = k x є RDomain f = R Range f = {k}Graph: The graph of a constant

function is a line parallel to x-axis.

x 1 -1 0

y k k k

k є R

2

4

6

8

-2-4-6-8 2 4 6 8-2

-4

-6

-8

0

y = k (k = 3)

Y

X’ X

Y’

Some Real Functions and their GraphsSome Real Functions and their Graphs

2. Identity function

Def: A function f : R R is called a identity function if f maps every element of R to itself.

Rule: f (x) = x x є R

Domain f = R

Range f = R

Graph: The graph of a identity function is a line passing through the origin. It lies in the first and the third quadrants where x and y take the same sign

x 1 -1 0

y 1 -1 0

2

4

6

8

-2-4-6-8 2 4 6 8-2

-4

-6

-8

0

y = x

Y

X’ X

Y’

Some Real Functions and their GraphsSome Real Functions and their Graphs

3. The Modulus function Def: A function f : R R is called a

modulus function if f maps every element x of R to its absolute value.

Rule: f (x) = |x| x є R. Where x when x 0

| x | =-x when x < 0

Domain f = R Range f = [0, )Graph: The graph of a modulus

function is a V shaped function lying above the x-axis. It passes through the origin.

x 1 -1 0

y 1 1 0

2

4

6

8

-2-4-6-8 2 4 6 8-2

-4

-6

-8

0

y = |x|

Y

X’ X

Y’

Some Real Functions and their GraphsSome Real Functions and their Graphs

4. Polynomial function

Def: A function f : R R is called a polynomial function if f maps every element x of R to a polynomial in x

Rule: f (x) = ax2 + bx + c x є R. (it can be a polynomial of any degree)

Domain f = R

Range f = R

Graph: The graph of a quadratic function is a parabola

x 1 -1 0

y 1 1 0

y = x2

2

4

6

8

-2-4-6-8 2 4 6 8-2

-4

-6

-8

0

y = x2

Y

X’ X

Y’

Some Real Functions and their GraphsSome Real Functions and their Graphs

5. Rational function Def: A function f : R R is called a

rational function if f maps every element x of R to a rational function in x

Rule: f(x) = h(x) g(x) where h(x) and

g(x) are polynomial functions of x defined in the domain and g(x)0

Domain f = R- {roots of g(x)}Range f = RGraph: The graph of a rational

function varies from function to function.

Y = 1/x

x -2 -1.5 -1

-0.5 0.25 0.5 1 1.5 2

y -0.5 -0.67 -1

-2 4 2 1 0.67 0.5

Some Real Functions and their GraphsSome Real Functions and their Graphs

6. Signum function Def: A function f : R R is called a

signum function if f maps every element x of R to the {-1,0,1} of the co-domain R.

Rule: 1, if x > 0f (x) = 0, if x = 0

-1,if x < 0Domain f = RRange f = {-1,0,1} Graph: The graph of the signum

function corresponds the graph of the function | x |

f (x) = x

x -3 -2 -1 0 1 2 3 4 5

y -1 -1 -1 0 1 1 1 1 1

y = | x | / x

Some Real Functions and their GraphsSome Real Functions and their Graphs7. Greatest Integer function Def: A function f : R R is called a greatest integer function if f maps every element x of R to the greatest integer which is less than or equal to x. Rule: f (x) = [x], x є RTo find [1] = the greatest of all the integers which are 1…….. -3, -1, 0, 1 are the integers which are 1.of these 1 is the greatest integer.[-2.5] = -3Domain f = RRange f = ZGraph: The graph of the greatest integer function suggest another name for this function as step function.

x -4x< -3 -1x < 0 0x <1 3x <4

y -4 -1 0 3

y = [x]

REFERENCEREFERENCE

• 1.NCERT TEXT BOOK CLASS XI

• 2.MATHEMATICS CLASS XI BY

• R.D.SHARMA

• 3. www.en.wikipedia.org