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 єA, 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

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 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

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 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)}

6. 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 2. Arrow Diagram 1. Tabular Diagram for R A A A R A

7. B A f x y = f(x) 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(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.

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. B A f X y z 1 2 3 4 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. A f B 2 3 4 1 3 6 8 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. X f Y 3 6 8 a b c 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. X f F 1 5 7 2 3 4 7 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. X g Y 2 3 4 7 1 2 3 4 5 6 7 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 є R is a real function.

15. Y 8 6 y = k (k = 3) 4 2 X X’ 0 -8 -6 -4 -2 2 4 6 8 -2 -4 -6 -8 Y’ 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 є R such that f(x) = k  x є R Rule: f(x) = k  x є R Domain f = R Range f = {k} Graph: The graph of a constant function is a line parallel to x-axis. k є R

16. Y 8 6 y = x 4 2 X X’ 0 -8 -6 -4 -2 2 4 6 8 -2 -4 -6 -8 Y’ Some Real Functions and their Graphs 2. Identityfunction 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

17. Some Real Functions and their Graphs 3. TheModulusfunction 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. Y 8 y = |x| 6 4 2 X X’ 0 -8 -6 -4 -2 2 4 6 8 -2 -4 -6 -8 Y’

18. Y 8 6 y = x2 4 2 X X’ 0 -8 -6 -4 -2 2 4 6 8 -2 -4 -6 -8 Y’ Some Real Functions and their Graphs y = x2 4. Polynomialfunction 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

19. Some Real Functions and their Graphs 5. Rationalfunction 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

20. Some Real Functions and their Graphs y = | x | / x 6. Signumfunction 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 > 0 f (x) = 0, if x = 0 -1,if x < 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

21. Some Real Functions and their Graphs y = [x] 7. Greatest Integerfunction 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 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.

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