CLASS12 RELATIONS AND FUNCTIONS

Transcription

1 CLASS12 RELATIONS AND FUNCTIONS Deepak sir Relations and functions, domain, co-domain and range have been introduced in Class XI along with different types of specific real valued functions and their graphs. 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. M A S T E R S T U I T I O N C E N T E R 113- F, A R J U N N A G A R, S J E N e w D e l h i

2 RELATIONS AND FUNCTIONS Introduction Recall that the notion of relations and functions, domain, co-domain and range have been introduced in Class XI along with different types of specific real valued functions and their graphs. 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. Types of Relations In this section, we would like to study different types of relations. We know that a relation in a set A is a subset of A X A. Thus, the empty set φ and A x A are two extreme relations. For illustration, consider a relation R in the set A = {1, 2, 3, 4} given by R = {(a, b): a - b = 10}. This is the empty set, as no pair (a, b) satisfies the condition a - b = 10. Similarly, R' = {(a, b) : a - b > 0} is the whole set A x A, as all pairs (a, b) in A x A satisfy a - b > 0. These two extreme examples lead us to the following definitions. EMPTY RELATION : A relation R in a set A is called empty relation, if no element of A is related to any element of A, i.e., R = ϕ ϵ A x A. UNIVERSAL RELATION : A relation R in a set A is called universal relation, if each element of A is related to every element of A, i.e., R = A x A. Both the empty relation and the universal relation are some times called trivial relations. Q1. Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a, b) : a is sister of b} is the empty relation and R' = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation. Solution Since the school is boys school, no student of the school can be sister of any student of the school. Hence, R = showing that R is the empty relation. It is also obvious that the difference between heights of any two students of the school has to be less than 3 meters. This shows that R' = A x A is the universal relation Note: In Class XI, we have seen two ways of representing a relation, namely raster method and set builder method. However, a relation R in the set {1, 2, 3, 4} defined by R = {(a, b) : b = a + 1} is also expressed as a R b if and only if b = a + 1 by many authors. We may also use this notation, as and when convenient. If (a, b) ϵ R, we say that a is related a to b and we denote it as a R b. Consider three types of relations, 1 P a g e

3 REFLEXIVE, SYMMETRIC AND TRANSITIVE. REFLEXIVE RELATION : A relation R in set A is said to be reflexive, if (a,a) ϵr, for every a ϵ A or we say ara, for every aϵa. SYMMETRIC RELATION : A relation R in set A is said to be symmetric, if (a, b) ϵ R (b, a) ϵr, for all a, bϵ A. We can also say arb for every a,b e A. TANSITIVE RELATION : A relation R in set A is said to be transitive, if (a, b) ϵ R and (b, c) ϵ R (a, c) ϵ R, for every a,b,c ϵ R. We can say arb, brc,then arb for all a,b,c ϵ R. A relation R in set A is said to be an equivalence relation if relation R is reflexive, symmetric and transitive. A. Let A = { 1,2,3} then R1={(1,1),(2,2),(3,3),(1,2),(1,3)} is reflexive and R2 = {(1,1),(3,3),(1,2),(1,3)}it is not reflexive as (2,2) R. B. Let A = { a,b,c} then R1={(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,a),(c,b) (c,c)} is reflexive and R2={(a,a),(a,b),(a,c),(b,b),(b,c),(c,a),(c,b) (c,c)} reflexive, (b,a) R2 non symmetric R3={(a,a),(a,b),(a,c),(b,b),(b,c),(c,a),(c,b) (c,c)} reflexive, (b,a) R3 non symmetric non transitive as we have (b,c ), (c,a) but no (b,a). R4={(a,a),(a,b),(b,c),(c,a),(c,b) (c,c)} R5={(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,b) (c,c)} Example Show that the relation R in the set Z of integers given by R = {(a, b) : 2 divides a - b} is an equivalence relation. Solution : R is reflexive, as 2 divides (a - a) for all a ϵ Z. Further, if (a, b) ϵ R, then 2 divides a - b. Therefore, 2 divides b - a. Hence, (b, a)ϵ R, which shows that R is symmetric. Similarly, if (a, b) ϵ R and (b, c) e R, then a - b and b - c are divisible by 2. Now, a - c = (a - b) + (b - c) is even. So, (a - c) is divisible by 2. This shows that R is transitive. Thus, R is an equivalence relation in Z. Example Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} R = {(a, b) : both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other and all the elements of the subset {2, 4, 6} are related to each other, but no element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}. Solution Given any element a in A, both a and a must be either odd or even, so that (a, a) e R. Further, (a, b) ϵ R both a and b must be either odd or even (b, a) ϵ R. Similarly, (a, b) ϵ R and (b, c) ϵ R all elements a, b, c, must be either even or odd simultaneously (a, c) ϵ R. Hence, R is an equivalence relation. Further, all the elements of {1, 3, 5, 7} are related to each other, as all the elements of this subset are odd. Similarly, all the elements of the subset {2, 4, 6} are related to each other, as all of them are even. Also, no element of the subset {1, 3, 5, 7} can be related to any element of {2, 4, 6}, as elements of {1, 3, 5, 7} are odd, while elements of {2, 4, 6} are even. 2 P a g e

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21 Types of Functions The notion of a function along with some special functions like identity function, constant function, polynomial function, rational function, modulus function, signum function etc. along with their graphs have been given in Class XI. A function f is a rule from set A to set B which assigns to each element of set A, a unique element of set B. Set A is called the DOMAIN of the function 'f' and the set B is known as its CO-DOMAIN. The SET OF VALUES, from the set B which are ACTUALLY TAKEN by the function 'f' is called the RANGE OF THE FUNCTION 'f'. We denote it as f: A B, if x ϵ A then f(x)b. A function whose domain and co-domain are the sets of real numbers is known as a real valued function i.e. f: R R. IDENTITY FUNCTION. A function 'f' is said to be an identity function, if f(x) = x, x ϵ R. Df = Real numbers, R f = Real Number. CONSTANT FUNCTION. A function 'f' is said to be an constant function, if f(x) = c, x ϵ R. D f =R, R f = {c}. Positive square root function: A function f(x)= x, x>0is known as a positive square root function. D f =R + {0}, R f = R + {0}. Rational function. A function f(x) = p(x), where p(x) and q(x) are polynomial functions and q(x) q(x) 0. D f = R - {zeros of q(x)}. Absolute value function. A function f(x) = IxI, xϵ R is known as an absolute value function and is defined as f(x) = { x x > 0, f(x) is always positive. x x 0 D f = Real number ; R f = R + {0}. Greatest integer function. A function(x) = [x], xϵ R, where [x] represents greatest integer x, is called a greatest integer function, D f = Real number ; R f = Integers. Exponential function. A function f{x) = e x, xϵ R is known as an exponential function, 2 < e < 3. D f = Real Numbers,R f = R P a g e

22 Logarithmic function. A function fix) = log e x, x > 0 or f(x) = ln(x), x > 0 is known as logarithmic function. D f = Real Numbers, R f = R +. Properties of logarithm: a) loga a = 1 a b = M, then logam=b b) loga 1 = 0 as a 0 = 1 c) logm ab = logm a + logmb provided a and b are both positive, i.e., a, b> 0 d) logm { a }= logm a - logm b provided a and b are both positive, i.e., a,b> 0 b e) logm (a) n = n logm a provided a and b are both positive, i.e., a, n > 0 An even function. A function 'f' is said to be an even function, if f(-x) =f(x). An odd function. A function '/'is said to be an odd function, if f(- x) = -f(x). Algebra of functions. If 'f' and 'g' are two given functions with domains D l and D 2, then (1) (f± g) (x) = f(x) ±g(x) (2) (fg)(x) = f(x).g(x) Domain of f± g,fg is D x D v Domain of f g is D l D 2 - {zeros of g(x)}. One-One or injective function. A function f: A B is said to be one-one (or injective), if the images of distinct element of A under the rule '/' are distinct in B, i.e., for every a, bϵ A, a b f(a) f(b) or we can also say that f(a) =f(b) a=b. One one function Many-one function. A function f: A B is said to be many-one, if the images of two or more elements of A are associated with same elements of B, i.e., for everya, bϵ A, a b f(a) = f(b). 21 P a g e

23 Many one function : when function is not one-one then many one. For any set X and any subset S of X the inclusion map S X (which sends any element s of S to itself) is injective. In particular the identity function X X is always injective (and in fact bijective). If the domain X = or X has only one element, the function X Y is always injective. The function f : R R defined by f(x) = 2x + 1 is injective. The function g : R R defined by g(x) = x 2 is not injective, because (for example)g(1) = 1 = g( 1). However, if g is redefined so that its domain is the non-negative real numbers [0,+ ), then g is injective. The exponential function exp : R R defined by exp(x) = e x is injective (but notsurjective as no real value maps to a negative number). The natural logarithm function ln : (0, ) R defined by x ln x is injective. The function g : R R defined by g(x) = x n x is not injective, since, for example,g(0) = g(1). Onto or surjective function. A function f : A B is said to be onto function (or suijective), if every element of B is image of some element of A under the rule 'f', i.e., for every be B, there exists an element a ϵ A such that f(a) = b. Note : A function is onto if and only if R f = B. On to The function f3 and f4 are onto as every element has it image 22 P a g e

24 Into-function. A function f: A B is said to be an into function if there is at least one element in B which is not an image of any element in A, i.e., Range Codomain. Also there is at least one b ϵ B such that b f(a), for any a ϵ A. the function f in Fig 1.2 (i) is not onto as elements e, f in X2 are not the image of any element in X1 under. One-one onto function. A function: A > B is said to be one-one onto (or bijective) if 'f' is both one- one and onto. IF FUNCTION IS NOT ONTO THEN INTO. For a pairing between X and Y (where Y need not be different from X) to be a bijection, four properties must hold: 1. each element of X must be paired with at least one element of Y, 2. no element of X may be paired with more than one element of Y, 3. each element of Y must be paired with at least one element of X, and 4. no element of Y may be paired with more than one element of X. Satisfying properties (1) and (2) means that a bijection is a function with domain X. It is more common to see properties (1) and (2) written as a single statement: Every element of X is paired with exactly one element of Y. Functions which satisfy property (3) are said to be "onto Y " and are called surjections (or surjective functions). Functions which satisfy property (4) are said to be "one-to-one functions" and are called injections (or injective functions).[1] With this terminology, a bijection is a function which is both a surjection and an injection, or using other words, a bijection is a function which is both "one-to-one" and "onto". For any set X, the identity function id X on X is surjective. The function f : Z {0,1} defined by f(n) = n mod 2 (that is, even integers are mapped to 0 and odd integers to 1) is surjective. The function f : R R defined by f(x) = 2x + 1 is surjective (and even bijective), because for every real number y we have an x such that f(x) = y: an appropriate x is (y 1)/2. 23 P a g e

25 The function f : R R defined by f(x) = x 3 3x is surjective, because the pre-image of any real number y is the solution set of the cubic polynomial equation x 3 3x y = 0 and every cubic polynomial with real coefficients has at least one real root. However, this function is not injective (and hence not bijective) since e.g. the pre-image of y = 2 is {x = 1, x = 2}. (In fact, the pre-image of this function for every y, 2 y 2 has more than one element.) The function g : R R defined by g(x) = x 2 is not surjective, because there is no real number x such that x 2 = 1. However, the function g : R R0 + defined by g(x) = x 2 (with restricted codomain) is surjective because for every y in the nonnegative real codomain Y there is at least one x in the real domain X such that x 2 = y. The natural logarithm function ln : (0,+ ) R is a surjective and even bijective mapping from the set of positive real numbers to the set of all real numbers. Its inverse, the exponential function, is not surjective as its range is the set of positive real numbers and its domain is usually defined to be the set of all real numbers. HOW TO SOLVE ONE-ONE Step1. Take two arbitrary elementes x, y in the domain of f. Step2.a. Put f(x) = f(y) If f(x) = f(y) give x = y only then f: A B is a one one function. b.let x1 x2 then if f(x1) f(x2) If function is not one one then many one. How to solve onto Step 1: Take two arbitrary elementes y in B. Step 2: Put f(x) = y Step 3:Solve the equation f(x) =y for x and obtain x in tearm of y. let x=h(y) Step 4: if for all values of yϵ B, the value of x obtained from x=h(y) are in A,then f is onto. if for all values of yϵ B, the value of x obtained from x=h(y) are not in A,then f is onto. 24 P a g e

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34 Composition of Functions and Invertible Function Definition 8 Let f: A B and g : B C be two functions. Then the composition of f and g, denoted by gof, is defined as the function gof: A C given by gof (x) = g(f (x)), y x e A. It can be verified in general that gof is one-one implies that f is one-one. Similarly, gof is onto implies that g is onto. The interesting fact is that the result mentioned in the above example is true for an arbitrary one-one and onto function f: X Y. Not only this, even the converse is also true, i.e., iff: X ^ Y is a function such that there exists a function g : Y X such that gof = Ix and fog = Iy, then f must be one-one and onto. Composition of functions. Let f: A B and g : B C be two given functions. Then composition of function f and g is a function from A to C and is denoted as gof We define 'gof' as (gof) (x) = g {f (x)} x ϵ A. For working, on element x first we apply f rule and whatever result is obtained in set B, we apply g rule on it to get the required result in set c. Q Let S = {1, 2, 3}. Determine whether the functions f: S S defined as below have inverses. Find f1, if it exists. f = {(1, 1), (2, 2), (3, 3)} f = {(1, 2), (2, 1), (3, 1)} f = {(1, 3), (3, 2), (2, 1)} Solution It is easy to see thatf is one-one and onto, so thatf is invertible with the inverse f-1 off given by f-1 = {(1, 1), (2, 2), (3, 3)} = f Since f (2) = f (3) = 1, f is not one-one, so that f is not invertible. It is easy to see that f is one-one and onto, so that f is invertible with f-1 = {(3, 1), (2, 3), (1, 2)}. 33 P a g e

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52 Binary Operation A binary operation * on set A is said to be commutative, if for all a, & t rx, ct ' U u (X. A binary operation * on set A is said to be associative, if for all a, c e A, (a * b) * c - a * (b * c). Right from the school days, you must have come across four fundamental operations namely addition, subtraction, multiplication and division. The main feature of these operations is that given any two numbers a and b, we associate another number a + b or a - b or ab or a/b, b 0. It is to be noted that only two numbers can be added or multiplied at a time. When we need to add three numbers, we first add two numbers and the result is then added to the third number. Thus, addition, multiplication, subtraction and division are examples of binary operation, as 'binary' means two. If we want to have a general definition which can cover all these four operations, then the set of numbers is to be replaced by an arbitrary set X and then general binary operation 51 P a g e

53 is nothing but association of any pair of elements a, b from X to another element of X. This gives rise to a general definition as follows: Binary operation implies an operation involving the two elements of a given set. A binary operation * on set A is a function * : A x A A. We denote * (a, b) by a * b. Q1. Show that binary operation * : NXN N given by a* b = a + b + ab is Z valid operation. Sol.We notice that for a, b ϵ N, a + b is a natural number and ab is also a natural number. Hence a + b + ab ϵ N for all a, bϵ N. Therefore a * b is a valid binary operation in N. Ex. 2. Show that binary operation * : N XN N given by a * b = a + b - ab is not a valid operation. Sol. We notice that for a, b ϵ N, a + b is a natural number and ab is also a natural number but a+ b-ab may or may not be a natural number. e.g., let a = 3, b = 5 then, a*b = a + b-ab =>. 3*5 = x5 = 8-15 = -7 ϵ N Hence a* b is not a valid operation in N. 3 Show that the : R x R R given by (a, b) max {a, b} and the : R x R R given by (a, b) min {a, b} are binary operations. Solution Since carries each pair (a, b) in R x R to a unique element namely maximum of a and b lying in R, is a binary operation. Using the similar argument, one can say that a is also a binary operation. Remark (4, 7) = 7, (4, - 7) = 4, (4, 7) = 4 and (4, - 7) = - 7. When number of elements in a set A is small, we can express a binary operation * on the set A through a table called the operation table for the operation *. For example consider A = {1, 2, 3}. A binary operation on the set X is called commutative, if a b= b a, for every a, b X. Show that + : R x R R and X : R x R R are commutative binary operations, but - : R x R R and : R* x R* R* are not commutative. ExShow that * : R x R ^ R defined by a * b = a + 2b is not commutative. If we want to associate three elements of a set X through a binary operation on X, we encounter a natural problem. The expression a * b * c may be interpreted as (a * b) * c or a * (b * c) and these two expressions need not be same. For example, (8-5) - 2 * 8 - (5-2). Therefore, association of three numbers 8, 5 and 3 through the binary operation 'subtraction' is meaningless, unless bracket is used. But in case of addition, has the same value whether we look at it as ( 8 + 5) + 2 or as 8 + (5 + 2). Thus, 52 P a g e

54 association of 3 or even more than 3 numbers through addition is meaningful without using bracket. This leads to the following: A binary operation * : A x A A is said to be associative if (a * b) * c = a * (b * c), a, b, c, ϵ A. Ex.Show that addition and multiplication are associative binary operation on R. But subtraction is not associative on R. Division is not associative on R*. Example 37 Show that * : R x R R given by a * b= a + 2b is not associative. Remark Associative property of a binary operation is very important in the sense that with this property of a binary operation, we can write a1 * a2 *... * a n which is not ambiguous. But in absence of this property, the expression a1 * a2 *... * a n is ambiguous unless brackets are used. Recall that in the earlier classes brackets were used whenever subtraction or division operations or more than one operation occurred. For the binary operation '+' on R, the interesting feature of the number zero is that a + 0 = a = 0 + a, i.e., any number remains unaltered by adding zero. But in case of multiplication, the number 1 plays this role, as a x 1 = a = 1 x a, a in R. This leads to the following definition: D3 Given a binary operation * : A x A A, an element e ϵ A, if it exists, is called identity for the operation *, if a * e = a = e * a, a ϵ A. Ex Show that zero is the identity for addition on R and 1 is the identity for multiplication on R. But there is no identity element for the operations - : R x R ^ R and : R* x R* R*. Solution a + 0 = 0 + a = a and a x 1 = a = 1 x a, V a e R implies that 0 and 1 are identity elements for the operations '+' and ' x' respectively. Further, there is no element e in R with a - e = e - a, v a. Similarly, we can not find any element e in R* such that a - e = e - a, V a in R*. Hence, '-' and '-' do not have identity element. Remark Zero is identity for the addition operation on R but it is not identity for the addition operation on N, as 0 N. In fact the addition operation on N does not have any identity. One further notices that for the addition operation + : R x R R, given any a ϵ R, there exists - a in R such that a + (- a) = 0 (identity for '+') = (- a) + a. Similarly, for the multiplication operation on R, given any a 0 in R, we can choose 1/a in R such that a X 1/a = 1 (identy for X). Given a binary operation : A A A with the identity element e in A, an element a A is said to be invertible with respect to the operation, if there exists an element bin A such that a b= e= b a and b is called the inverse of a and is denoted by a 1. Example 39 Show that ais the inverse of afor the addition operation + on R and 1/a is the inverse of a 0 for the multiplication operation on R. 53 P a g e

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94 1) Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive. 2) Show that the relation R in the set Z of integers given by R = {(a, b) : 2 divides a - b} is an equivalence relation. Determine whether each of the following relations are reflexive, symmetric and transitive: 3) Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4,4),(1, 3), (3, 3), (3, 2)}. 4) Relation R in the set A = {1, 2, 3,..., 13, 14} defined as R = {(x, y) : 3x - y = 0} 5) Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4} 6) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x,y) : y is divisible by x} 7) Relation R in the set Z of all integers defined as R = {(x, y) : x - y is an integer} Relation R in the set A of human beings in a town at a particular time given by 8) R = {(x, y) : x and y work at the same place} 9) R = {(x, y) : x and y live in the same locality} 10) R = {(x, y) : x is exactly 7 cm taller than y} 11) R = {(x, y) : x is wife of y} 12) R = {(x, y) : x is father of y} 13) Show that the relation R in the set R of real numbers, defined as R = {(a, b) : a < b 2 } is neither reflexive nor symmetric nor transitive. {reflexisive(1/2,1/2), non symmetric for (1,-4),non transitive for (2,-3) } 14) Show that the relation R in the set R of real numbers, defined as R = {(a, b) : a b } is reflexive, transitive but not symmetric. 15) Show that the relation R in R defined as R = {(a, b) : a < b}, is reflexive and transitive but not symmetric. 16) Check whether the relation R in R defined by R = {(a, b) : a < b3} is reflexive, symmetric or transitive. 17) Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive. 18) Let T be the set of all triangles in a plane with R a relation in T given by R = {(T1, T2) : T1 is congruent to T2}. Show that R is an equivalence relation. 19) Let T be the set of all triangles in a plane with R a relation in T given by R = {(T1, T2) : T1 is similer to T2}. Show that R is an equivalence relation. 20) Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L1, L2) : L1 is perpendicular to L2}. Show that R is symmetric but neither reflexive nor transitive. 21) Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L1, L2) : L1 is parllel to L2}.check for equalance relation. 22) Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1 is similar to T2}, is equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related? 23) Show that the relation R defined in the set A of all polygons as R = {(P1, P2) : P1 and P2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5? 24) Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2) : L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x P a g e

95 25) Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b) : both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other and all the elements of the subset {2, 4, 6} are related to each other, but no element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}. 26) Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive. 27) Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : a - b is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}. 28) Show that each of the relation R in the set A = {x e Z : 0 < x < 12}, given by A. R = {(a, b) : a - b is a multiple of 4} B. R = {(a, b) : a = b} is an equivalence relation. Find the set of all elements related to 1 in each case. 29) Show that the relation R in the set A of points in a plane given by R = {(P, Q) : distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all points related to a point P (0, 0) is the circle passing through P with origin as centre. 30) Prove the following relation R on the set N X N defined by (a,b) R(c,d) a+d = b+c for all (a,b),(c,d) ϵ N X N. is an equalance relation. (a,b) R(c,d) ad= bc for all (a,b),(c,d) ϵ N X N. is an equalance relation. (a,b) R(c,d) ad(b+c) = bc(a+d) for all (a,b),(c,d) ϵ N X N. is an equalance relation. 1) Show that the function f: N N, given by f (x) = 2x, is one-one but not onto. 2) Prove that the function f: R R, given by f(x) = 2x, is one-one and onto. 3) Show that the function f: R R, defined as f(x) = x 2, is neither one-one nor onto..check the injectivity and surjectivity of the following functions: 4) f: N N given by f(x) = x 2 5) f: Z Z given by f (x) = x 2 6) f :R R given by f (x) = x 2 7) f :N N given by f (x) = x 3 8) f: Z Z given by f (x) = x 3 9) Prove that the Greatest Integer Function f: R R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x. 10) Show that the Modulus Function f: R R, given by f(x) = x, is neither one- one nor onto, where x is x, if x is positive or 0 and x is - x, if x is negative. 11) In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. a. f: R R defined by f(x) = 3-4x 94 P a g e

96 b. f: R R defined by f(x) = 1 + x 2 12) Let A and B be sets. Show that f: A x B B x A such that f (a, b) = (b, a) is bijective function. 13) Let f : R R be defined as fx) = x 4. Choose the correct answer. (A) f is one-one onto (B) f is many-one onto (C) f is one-one but not onto (D) f is neither one-one nor onto. 14) Let f : R R be defined as f(x) = 3x. Choose the correct answer. (A) f is one-one onto (B) f is many-one onto (C) f is one-one but not onto (D) f is neither one-one nor onto. 1) Let f: N Y be a function defined as f (x) = 4x + 3, where, Y = {y e N: y = 4x + 3 for some x e N }. Show that f is invertible. Find the inverse. 2) Let Y = {n2 : n e N} c N. Consider f: N Y as f (n) = n2. Show that f is invertible. Find the inverse off 3) Let f: N R be a function defined as f(x) = 4x x Show that f: N S, where, S is the range off is invertible. Find the inverse off 4) Let f: {1, 3, 4} {1, 2, 5} and g : {1, 2, 5} {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof. 5) Shoe that if f: R -{ 7 } R 5 -{3 } is defined by f(x) =3x 4 and f: R 5 5x 7 -{3} R 5 -{7 } is defined by 5 f(x) = 7x 4 then fog = IA and gof = IB where, A= R 5x 3 -{3}, B= R 5 -{7 } ; IA(x)=x,xϵA, IB(x)=x 5,xϵB are called identity fuction A and B, respectively. State with reason whether following functions have inverse 6) f: {1, 2, 3, 4} {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)} 7) g : {5, 6, 7, 8} {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)} 8) h : {2, 3, 4, 5} {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)} 9) Consider f: R R given by f (x) = 4x + 3. Show that f is invertible. Find the inverse of f. 10) Consider f: R+ [4, ) given by f(x) = x Show that f is invertible with the inversef1 off given byf -1 (y) = y 4, where R, is the set of all non-negative real numbers. 11) Consider f: R+ [- 5, ) given by f (x) = 9x 2 + 6x - 5. Show that f is invertible 12) Iff: R R be given by f(x) = (3 - x3) 1/3, then fof (x) is Determine whether or not each of the definition of * given below gives a binary operation. In the event that * is not a binary operation, give justification for this. 1) On Z+, define * by a * b = a - b 2) On Z+, define * by a * b = ab 3) On R, define * by a * b = ab2 4) On Z+, define * by a * b = a - b 5) On Z+, define * by a * b = a 95 P a g e

97 For each operation * defined below, determine whether * is binary, commutative or associative. 6) On Z, define a * b = a - b 7) On Q, define a * b = ab + 1 8) On Q, define a * b = 9) On Z+, define a * b = 2ab 10) On Z+, define a * b = ab 11) On R - {- 1}, define a * b = 12) Consider the binary operation on the set {1, 2, 3, 4, 5} defined by a b = min {a, b}. Write the operation table of the operation a. 13) Consider a binary operation on the set {1, 2, 3, 4, 5} given by the following a. Compute (2 * 3) * 4 and 2 * (3 * 4) b. Is * commutative? c. Compute (2 * 3) * (4 * 5). 14) Let *' be the binary operation on the set {1, 2, 3, 4, 5} defined by a *' b = H.C.F. of a and b. Is the operation *' same as the operation * defined in Exercise 4 above? Justify your answer. 15) Let * be the binary operation on N given by a * b = L.C.M. of a and b. a. Find 5 * 7, 20 * 16 b. Is * commutative? c. Is * associative? d. Find the identity of * in N e. Which elements of N are invertible for the operation *? Is * defined on the set {1, 2, 3, 4, 5} by a * b = L.C.M. of a and b a binary operation? Justify your answer. 16) Let * be the binary operation on N defined by a * b = H.C.F. of a and b. Is * commutative? Is * associative? Does there exist identity for this binary operation on N? 17) Let * be a binary operation on the set Q of rational numbers as follows: a. a * b = a - b b. a * b = a2 + b2 c. a * b = a + ab d. a * b = (a - b) 2 e. a * b = ab /4 f. a * b = ab 2 Find which of the binary operations are commutative and which are associative. 18) Find which of the operations given above has identity. 19) Let A = N x N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d) 20) Show that * is commutative and associative. Find the identity element for * on A, if any. 21) State whether the following statements are true or false. Justify. a. For an arbitrary binary operation * on a set N, a * a = a V a e N. b. If * is a commutative binary operation on N, then a * (b * c) = (c * b) * a c. Consider a binary operation * on N defined as a * b = a3 + b3. Choose the correct answer. d. Is * both associative and commutative? e. Is * commutative but not associative? f. Is * associative but not commutative? 96 P a g e

98 g. Is * neither commutative nor associative? 22) Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Let R1 be a relation in X given by R1 = {(x, y) : x - y is divisible by 3} and R2 be another relation on X given by R2 = {(x, y): {x, y} c {1, 4, 7}} or {x, y} c {2, 5, 8} or {x, y} c {3, 6, 9}}. Show that R1= R2 23) Let f: R R be defined as f (x) = 10x + 7. Find the function g : R R such that gof=fog= R. 24) Let f: W W be defined as f (n) = n - 1, if n is odd and f (n) = n + 1, if n is even. Show that f is invertible. Find the inverse off. Here, W is the set of all whole numbers. 25) Iff: R R is defined by f(x) = x 2-3x + 2, find f (f(x)). 97 P a g e