LESSON RELATIONS & FUNCTION THEORY

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Philip Cannon
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2 2 Definitions LESSON RELATIONS & FUNCTION THEORY Ordered Pair Ordered pair of elements taken from any two sets P and Q is a pair of elements written in small brackets and grouped together in a particular order, i.e., (p,q), p P and q Q. Cartesian Product Given two non-empty sets P and Q. The cartesian product P Q is the set of all ordered pairs of elements from P and Q, i.e., P Q = { (p,q) : p P, q Q } If either P or Q is the null set, then P Q will also be empty set, i.e., P Q = Note: (i) Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal. (ii) If there are p elements in A and q elements in B, then there will be pq elements in A B, i.e., if n(a) = p and n(b) = q, then n(a B) = pq. (iii) If A and B are non-empty sets and either A or B is an infinite set, then so is A B. (iv) A A A = {(a, b, c) : a, b, c A}. Here (a, b, c) is called an ordered triplet. Relations The concept of the term relation in mathematics has been drawn from the meaning of relation in English language, according to which two objects or quantities are related if there is a recognisable connection or link between the two objects or quantities. A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product 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 second element is called the image of the first element. Domain of a Relation 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. Range of a Relation 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. Note that range codomain. Note: The total number of relations that can be defined from a set A to a set B is the number of possible subsets of A B. If n(a ) = p and n(b) = q, then n (A B) = pq and the total number of relations is 2 pq.

3 Functions Introduction If to every value (considered as real unless otherwise stated) of a variable x, which belongs to some collection (Set) E, there corresponds one and only one finite value of

3 3 Functions Introduction If to every value (considered as real unless otherwise stated) of a variable x, which belongs to some collection (Set) E, there corresponds one and only one finite value of the quantity y, then y is said to be a function (single valued) of x or a dependent variable defined on the set E; x is the argument or independent variable. If to every value of x belonging to some set E there corresponds one or several values of the variable y, then y is called a multiple valued function of x defined on E. Conventionally the word Function is used only as the meaning of single valued function, unless otherwise stated. Domain, Co-domain and Range Let f: A B, then the set A is known as the domain of f and the set B is known as co-domain of f. The set of all f images of elements of A is known as the range of f. Thus, Domain of f = {f(a) aa, (a, f(a)) f}, Range of f = {f(a) a A, f(a) B} It should be noted that range is a subset of co-domain. If only the rule of function is given then the domain of the function is the set of those real numbers, where function is defined. Some frequently used real functions Constant function A function f:a B, A, B R, is said to be a constant function if there exists a real number k such that f(x) = k, for all x A k Domain: A; Range: Notes for graph of f(x) = k: (i) The graph is a line parallel to x-axis. (ii) If k > 0, the graph will be a line above the x-axis and parallel to it. (iii) If k = 0, then the graph coincides with the x-axis. (iv) If k < 0, then the graph will be a line below the x-axis and parallel to it. Identity function The function from R to R that associates to each x R the same x, is called the identity function and is usually denoted by I. More precisely, we write I(x) = x, for all x R Domain: R; Range: R Notes for graph y = x: (i) The graph is a straight line (ii) It passes through the origin. (iii) Its slope is 1. Polynomial function A function f: R R is said to be a polynomial function if for each x in R, f(x) is a polynomial in x. The functions f (x) x x 2, g(x) x 2x

4 4 3 are some examples of polynomial functions whereas the function h(x) x 2x is not a polynomial function. Domain: R; Range: Depends on the polynomial representing the function. Notes for graph y = x 2 (A particular case): (i) The graph of y = x 2 is a parabola (ii) It is above the x-axis, except at one point, x = 0. (iii) It passes through the origin. (iv) It is symmetric with respect to the y-axis, sincef( x) f(x), for all x R. Modulus function The function f: R R defined by x, if x 0 f (x) x x, if x 0 is called the modulus function. It is also called absolute value function. Domain: R; Range: R 0 = {x: x is a non-negative real number} Notes for graph y = x : (i) The graph is symmetric with respect to the y-axis. (ii) It is above the x-axis, except at one point, x = 0. (iii) It passes through the origin. (iv) In the first quadrant, it coincides with the graph of the identity function. Square root function If x is a positive real number, then there are two square roots for x, of which one is positive. If we associate only the positive square root to any positive real number x, then we get a function f: R 0 R, called square root function and is given by f(x) = x. Domain: R 0 ; Range: R 0 : ( R 0 is set [0, )) Notes for graph y = x (i) The graph passes through the origin. (ii) It is above the x-axis except at one point, x = 0. (iii) As we move from left to right, the graph goes on rising above the x-axis, i.e., y increases as x increases. Signum Function The function f defined by 1, if x 0 f (x) 0, if x 0 1, if x 0 is called the signum function. Domain: R; Range:{ 1,0,1} 2 Greatest Integer function (Floor function)

5 5 The function f: R R defined by f(x) = [x] or x, x R is called the greatest integer function or the floor function. It is also called a step function. For example [4.2] = 4, [4.2] = 5, etc. Domain: R; Range: Z Notes for graph of y [x] (i) The graph consists of infinitely many broken pieces. (ii) Each piece of the graph coincides with the graph of a constant function. (iii) The graph lies within the first and third quadrants with the exception of one piece in [0, 1) which lies on x-axis. Smallest Integer Function (Ceiling function) The function f: R R defined by f (x) x, x R is called the Smallest integer function or the ceiling function. It is also a step function. Domain: R; Range: Z Notes for graph of y x (i) All the observations made for the floor function are also valid for this function (ii) The graph lies within the first and third quadrant with the exception of one piece in (1, 0] which lies on x-axis. Exponential function If a is a positive real number and a 1, then the function defined by x f (x) = a, x R is called an exponential function to the base a. Domain: R; Range: R +, the set of positive real numbers. x Notes for graph of y = 2 (A particular case) (i) The graph is above the x-axis. (ii) As we move from left to right, the graph goes on rising above the x-axis. (iii) The graph meets the y-axis at (0, 1). (iv) In the second quadrant, the graph approaches the x-axis but never meets it. Logarithmic function If a > 0 and a 1, then the function loga x y is called the logarithmic function (it is the inverse of the exponential function y to the base a), if and only if a x. Domain: R, the set of all positive real numbers; Range: R Notes for graph of y = log 2 x (A particular case) (i) The graph is on the right side of the y-axis. (ii) As we move from left to right, the graph goes on rising. (iii) The graph meets the x-axis at (1, 0).

6 (iv) In the fourth quadrant, the graph approaches to the y-axis but never touches it. 6 Trigonometric functions The domains and ranges of some of the trigonometric functions are as given below: Function (given by) Domain Range sin x R [1,1] cos x R [1,1] tan x R R (2n 1) : n Z 2 R n : n Z R cot x sec x (, 1] [1, ) R (2n 1) : n Z 2 R n : n Z (, 1] [1, ) cosec x Inverse trigonometric functions The domain and range (confining only to principal values) of inverse trigonometric functions are as given below: Function (given by) Domain Range (Principal Value) 1 sin x [1,1], cos x [1,1] [0, ] 1 tan x R, cot x R (0, ) 1 sec x (, 1] [1, ) 0,, cos ec x, 1][1, ),0 2 0, 2 CLASS XII Types of Relation In this section, we intend to define various types of relations of a given set A. 1. Empty/Void Relation Let A be a set. Then A A and so it is a relation on A. This relation is called the void or empty relation on A. 2. Universal Relation Let A be a set. Then A A A A and so it is a relation on A. This relation is called the universal relation on A. Note: The void and the universal relations on a set A are respectively the smallest and the largest relations of A.

7 7 3. Identity Relation Let A be a set. Then the relation I A = {(a, a) : a A} on A is called the identity relation on A. In other words, a relation I A on A is called the identity relation if every element of A is related to itself only. Example: The relation I A = {(1, 1), (2, 2), (3, 3)} is the identity relation on set A = {1, 2, 3}. But relations R 1 = {(1, 1), (2, 2)} and R 2 = {(1, 1), (2, 2), (3, 3), (1, 3)} are not identity relations on A, because (3, 3) R 1 and in R 2 element 1 is related to elements 1 and Reflexive Relation A relation R on a set A is said to be reflexive if every element of A is related to itself. Thus, R is reflexive (a, a) R for all a A. A relation R on a set A is not reflexive if there exists an element a A such that (a, a) R. Example: Let A = {1, 2, 3} be a set. Then R = {(1, 1), (2, 2), (3, 3), (1, 3), (2, 1)} is a reflexive relation on A. But R 1 = {(1, 1), (3, 3), (2, 1), (3, 2)} is not a reflexive relation on A, because 2 A but (2, 2) R 1. Example: A relation R on N defined by (x, y) R x y is a reflexive relation on N, because every natural number is greater than or equal to itself. Example: Let X be a non-void set and P(X) be the power set of X. A relation R on P(X) defined by (A, B) R A B is a reflexive relation since every set is subset to itself. Note 1: The identity relation on a non-void set A is always reflexive relation on A. However, a reflexive on A is not necessarily the identity relation on A. For example, the relation R = {(a, a), (b, b), (c, c), (a, b)} is a reflexive relation on set A = {a, b, c} but it is not the identity relation on A. Note 2: The universal relation on a non-void set A is reflexive. 5. Symmetric Relation A relation R on a set A is said to be a symmetric relation iff (a, b) R (b, a) R for all a, b A i.e. arb bra for all a, b A Example: Let L be the set of all lines in plane and let R be a relation defined on L by the rule (x, y) R x is perpendicular to y. Then R is symmetric relation on L, because L 1 L 2 L 2 L 1 i.e. (L 1, L 2 ) R (L 2, L 1 ) R. Example: Let S be a non-void set and R be a relation defined on power set P(S) by (A, B) R A B for all A, B P(S). Then R is not a symmetric relation. Example: Let A = {1, 2, 3, 4} and let R 1 and R 2 be relation on A given by R 1 = {(1, 3), (1, 4), (3, 1), (2, 2) (4, 1)} and R 2 = {(1, 1), (2, 2), (3, 3), (1, 3)}. Clearly, R 1 is a symmetric relation on A. However, R 2 is not so, because (1, 3) R 2 but (3, 1) R 2. Note 1: The identity and the Universal relations on a non-void set are symmetric relations. Note 2: A relation R on a set A is not a symmetric relation if there are at least two elements a, b A such that (a, b) R but (b, a) R. Note 3: A reflexive relation on a set A is necessarily symmetric. For example, the relation R= {(1, 1), (2, 2), (3, 3), (1, 3)} is a reflexive relation on set A = {1, 2, 3} but it is not symmetric. 6. Transitive Relation Let A be any set. A relation R on A is said to be a transitive relation iff (a, b) R and (b, c) R (a, c) R for all a, b, c A.

8 i.e. arb and brc arc for all a, b, c A. Note: The identity and the universal relations on a non-void set are transitive. 8 Example: The relation R on the set N of all natural numbers defined by (x, y) R x divides y, for all x, y N is transitive. Solution Let x, y, z N be such that (x, y) R and (y, z) R. Then (x, y) R and (y, z) R x divides y and y divides z there exists p, q N such that y = xp and z = yq z = (xp) q z = x(pq) x divides z [ pq N] (x, z) R Thus, (x, y) R, (y, z) R (x, z) R for all x, y, z N. Hence, R is a transitive relation on N. 7. Equivalence Relation A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive. Composite Function Let f: A B & g: B C be two functions. Then the function gof: A C defined by (gof) (x) = g (f(x)) x A is called the composite of the two functions f and g gof is defined if x A. f(x) is an element of the domain of g so that we can take its g-image. Properties of composite functions: (i) The composite of function is not commutative i.e. gof fog. (ii) The composite of functions is associative i.e. if f, g, h are three functions such that fo(goh) and (fog)oh are defined, then fo(goh) = (fog)oh. Kinds of Functions If f: A B is a function, then f associates all elements of set A to elements in set B such that an element of set A is associated to a unique element of set B. Following these two conditions we may associate different elements of set A to different elements of set B or more than one element of set A may be associated to the same element of set B. Similarly, there may be some elements in B which do not have their pre-images in A and all elements in B may have their pre-images in A. Corresponding to each of these possibilities we define some types of a function as given below. One-One Function (Injection) A function f: A B is said to be a one-one function or an injection if different elements of A have different images in B. Thus, f: A B is one one a b f(a) f(b) for all a, ba f(a) = f(b) a = b for all a, b A Note: Let f: A B and let x, y A. Then, x = y f(x) = f(y) is always true from the definition. But, f(x) = f(y) x = y is true only when f is one-one.

9 9 Many-One Function A function f: A B is said to be a many-one function if two or more elements of set A have the same image in B. Thus, f: A B is a many-one function if there exist x, y A such that x y but f(x) = f(y). In other words, f: A B is a many-one function if it is not a one-one function. Onto Function (Surjection) A function f: A B is said to be an onto function or a surjection if every element of B is the f- image of some element of A i.e., if f(a) = B or range of f is the co-domain of f. Thus, f: A B is a surjection iff for each b B, there exists a A such that f(a) = b. Into function A function f: A B is an into function if there exists an element in B having no pre-image in A. In other words, f: A B is an into function if it is not an onto function. Bijection (one-one onto function) A function f: A B is a bijection if it is one-one as well as onto. In other words, a function f: A B is a bijection if (i) It is one-one i.e. f(x) = f(y) x = y for all x, y A. (ii) It is onto i.e. for all y B, there exists x A such that f(x) = y. Inverse of a Function Let f: A B be a one-one and onto function, then there exists a unique function g: B A such that f(x) = y g(y) = x, x A & y B. Then g is said to be inverse of f. Thus g = f 1 : B A = {(f(x), x) (x, f(x)) f}. Properties of inverse function: (i) Inverse of a bijection is unique. (ii) If f: A B is a bijection and g: B A is the invese of f, then fog = I B and gof = I A, where I A & I B are identity functions on the sets A and B respectively. Note that the graph of f and g are the mirror images of each other on the line y = x. (iii) The inverse of an Identity function is the function itself, i.e. (f 1 ) 1 = f. (iv) The inverse of a bijection is also a bijection. (v) If f and g are two bijections f: A B, g: B C then the inverse of gof exists and (gof) 1 = f 1 og 1. Binary Operations Binary Operation: Let S be a non-empty set. A function f: S S S is called a binary operation on set S. 2 n If the finite set S has n elements, then total number of binary operations on S is n. Note for binary operations A binary operation on a set S associates each ordered pair (a, b) S S to a unique element f(a, b) in S. Instead of writing f (a, b) for the image of an orderd pair (a, b) S S, conventionally we will prefer to write a f b, that is we write f (a, b) as a f b.

10 10 Generally binary operations are denoted by the symbols *, 0, +, etc instead of the letters f, g, h etc. Thus, a binary operation *on a set S associates each ordered pair (a, b) in S S to a unique element a * b in S. Since an ordered pair is made of two elements of S. So, we can say that a binary operation * on a set S associates any two elements a, b of S to a unique element a * b in S. Types of Binary Operations 1. Commutativity: A binary operation on a set S is said to be a commutative binary operation, if a b = b a for all a, bs 2. Associativity: A binary operation on a set S is said to be an associative binary operation, if (a b) c = a (b c) for all a, bs 3. Distributivity: Let S be a non-empty set and and be two binary operations on S. Then, is said to be distributive over, if a (b c) = (a b) (a c) and, (b c) a = (b a) (c a) for all a, b, cs. Identity element: Let ''' be a binary operation on a set S. If there exists an element es such that a e = a = e a for all as. Then, e is called an identity element for the binary operation '' on set S. If S has an identity element for, then it is unique. Inverse of an element: Let be a binary operation on a set S, and let e be the identity element in S for the binary operation on S. Then, an element as is called an invertible element if there exists an element bs such that a b = e = b a The element b is called an inverse of element a. Addition Modulo n: Let n be a positive integer greater than 1 and a, bz n, where Z n = {0, 1, 2,.., (n 1)}. Then, we define addition modulo n i.e. + n as follows: a + n b = Least non-negative remainder when a + b is divided by n. Multiplication Modulo n: Let n be a positive integer greater than 1 and a, bz n, where Z n = {0, 1, 2, 3,, (n 1} Then, we define multiplication modulo n i.e., n as follows: a n b = Least non-negative remainder when ab is divided by n. Some useful methods for finding injectivity and surjectivity of a function Method to Check the Injectivity of a Function I Take two arbitrary elements x, y (say) in the domain of f. II Put f(x) = f(y) III Solve f (x) = f(y). If f(x) = f(y) gives x = y only, then f: A B is a one-one function (or an injection). Otherwise not. Method for Checking the Surjectivity of a Function Let f: A B be the given function. I Choose an arbitrary element y in B. II Put f(x) = y III Solve the equation f(x) = y for x and obtain x in terms of y. Let x = g(y)

11 IV If for all values of y B, the values of x obtained from x = g(y) are in A, then f is onto. If there are some y B for which x, given by x = g(y), is not in A. Then f is not onto. 11

Relations, Functions & Binary Operations

Relations, Functions & Binary Operations Important Terms, Definitions & Formulae 0 TYPES OF INTERVLS a) Open interval: If a and b be two real numbers such that a b then, the set of all the real numbers