CONTENT

1
CONTENT

• 1.ORDERED PAIRS
• 2.CARTESIAN PRODUCT OF SETS
• 3.RELATIONS
• 4.FUNCTIONS
• 5.ILLUSTRATIONS
• 6.REAL FUNCTIONS AND THEIR GRAPHS

2
Ordered 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.

3
Cartesian 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 ? A, b ? B. The Cartesian product
  of sets A and B is denoted by A x B. Thus AxB
  (a,b) a ? A and b ? B
• If A ? or B ?, then we define A x B ?
• 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

4
Relations

• 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 ? P, y ? 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

5
Number 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 ? A
• When x 1, 2x 1 3 ? A ? (1,3) ? R
• When x 2, 2x 1 5 ? A ? (2,5) ? 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)

6
2. Arrow Diagram
1. Tabular Diagram for R
A
A
A
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
R
A
7
Functions

• 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(?B) 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.

8

• Eg Let A 1,2,3,4 and B 1,6,8,11,15.
  Which of the following are functions from A to B?
• f A ? B defined by f(1) 1, f(2) 6, f(3)
  8, f(4) 8.
• f A ? B defined by f(1) 1, f(2) 6, f(3)
  15.
• f A ? B defined by f(1) 6, f(2) 6, f(3)
  6, f(4) 6.
• f A ? B defined by f(1) 1, f(2) 6, f(2)
  8, f(3) 8. f(4) 11.
• f A ? B defined by f(1) 1, f(2) 8, f(3)
  11, f(4) 15.

9
Pictorial 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.

10
Illustration 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

11
Illustration 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

12
Illustration 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

13
Illustration 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.

14
Real 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 ?
  R is a real function.

15
Some Real Functions and their Graphs

• Constant function
• Def A function f R ? R is called a constant
  function if there exists an element k ? R such
  that f(x) k ? x ? R
• Rule f(x) k ? x ? R
• Domain 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 ? R
16
Some 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 ? 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
17
Some Real Functions and their Graphs
x 1 -1 0
y 1 1 0

• 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 ? R. Where
• x when x ?? 0
• x
• -x when x lt 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.

Y
8
y x
6
4
2
X
X
0
-2
-4
-6
-8
2
4
6
8
-2
-4
-6
-8
Y
18
Some Real Functions and their Graphs
y x2

• 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 ? 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
19
Some 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 R
• Graph 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
20
Some Real Functions and their Graphs
y x / x

• 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 gt 0
• f (x) 0, if x 0
• -1,if x lt 0
• Domain f R
• Range 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
21
Some Real Functions and their Graphs
y x

• 7. 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 ? R
• To 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 -3
• Domain f R
• Range f Z
• Graph The graph of the greatest integer function
  suggest another name for this function as step
  function.

x -4?xlt -3 -1?x lt 0 0?x lt1 3?x lt4
y -4 -1 0 3
22
REFERENCE

• 1.NCERT TEXT BOOK CLASS XI
• 2.MATHEMATICS CLASS XI BY
• R.D.SHARMA
• 3. www.en.wikipedia.org