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RELATIONS AND, , FUNCTIONS, , , , BASIC CONCEPTS, , 1,, , Relation: If A and B are two non-empty sets, then any subset R of A ~ B is called relation from set A, to set B., , lity R:A>B&RCAxB, , If (x, y) € R, then we write x R y (read as x is R related to y) and if (x,y) ¢ R, then we write x Ry (read, as x is not R related to y)., , . Domain and Range of a Relation: If R is any relation from set A to set B then,, , (a) Domain of R is the set of all first coordinates of elements of R and itis denoted by Dom (R)., (b) Range of R is the set of all second coordinates of R and it is denoted by Range (R), , A relation R on set A means, the relation from A to A i.c., RGA x A., , . Some Standard ‘lypes of Relations:, , Let A be a non-empty set. Then, a relation R on set A is said to be, , (a) Reflexive: If (x, x) € R for each element x € A, i.c., if xRx for each element x € A., , (b) Symmetric: If (x, y) © R= (y, x) € R forall x,y € A, ie, if xRy => yRx forall x,y © A., , (c) Transitive: Lf (x, y) € Rand (y,z) € R= (x,z) € R forall x,y,z € A, ie, if xRy and yRz > xRz., , . Equivalence Relation: Any relation R ona set A is said to be an equivalence relation if R is reflexive,, , symmetric and transitive., , . Antisymmetric Relation: A relation R in a set A is antisymmetric, , if@beR (ba)eR > a=bVa,be R,oraRband bRa > a=b,Va,beER., For example, the relation “greater than or equal to, “2” is antisymmetric relation as, a2b,b2a >a=bV¥a,bv, , [Note: “Antisymmetric” is completely different from not symmetric.], , . Equivalence Class: Lel R be an equivalence relation on a non-emply set A. For all a € A, the, , equivalence class of ‘a’ is defined as the set of all such elements of A which are related to ‘a’ under, R_Itis denoted by [a], , ie., [a] = equivalence class of ‘a’ = {x € A: (x, a) € R}, , . Function: Let X and Y be two non-empty sets. Then, a rule f which associates to each element x € X, a, , unique element, denoted by f(x) of Y, is called 2 function from X to Y and written as f: X Y where,, [(x) is called image of x and x is called the pre-image of f(x) and the set Y is called the co-domain of, fand f(X) = {f(x): x € X} is called the range of f, , Relations and Functions | 5

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8., , 9., , 10., , i:, , 12., , 13,, , 14., , Types of Function:, , () One-one function (injective function): A function f : X > Y is defined to be one-one if the, image of distinct element of X under rule f are distinct, ic., for every x, X, € X, fx) = flr), implies that x, =X., , (ii) Onto function (Surjective function): A function f: X + Y is said to be onto function if each, element of Y is the image of some element of x i.c., for every y € Y, there exists some x € X, such, that y = f(x). Thus fis onto if range of f = co-domain of f., , (iii) One-one onto function (Bijective function): A function f : X > Y is said to be one-one onto, if, fis both one-one and onto., , (7) Many-one function: A function f : X — Y is said to be a many-one function if two or more, elements of set X have the same image in Y. i.c.,, , FX Yisa many-one function if there exist a, be X such thata #b but f(a) = f(b), , Composition of Functions: Let f: A B and g : B > C be two functions. Then, the composition of, fand g, denoted by gof, is defined as the function., fiAaB giBoCc, A B Cc, , gof:A>C given by, , Sof(x) = g(x), VE A, Clearly, dom(gof) = dom(f), Also, gof is defined only when range(f) C dom(g), Identity Function: Let R be the set of real numbers. A function I: R + R such that, , I(x) =x V x € Riscalled identity function., , Obviously, identity function associates each real number to itself., Invertible Function: For f: A — B, if there exists a function g : B > A such that gof = 1, and fog = Ip,, where I, and Ig are identity functions, then f is called an invertible function, and g is called the, inverse of f and it is written as f= g., , Number of Functions: If X and Y are two finite sets having m and n elements respectively then the, , number of functions from X to Y is 1”., , Vertical Line Test: It is used to check whether a relation is a function or not. Under this test, graph of, given relation is drawn assuming elements of domain along x-axis. Ifa vertical line drawn anywhere, in the graph, intersects the graph at only one point then the relation is a function, otherwise it is not, a function., , Horizontal Line Test: It is used to check whether a function is one-one or not. Under this test graph, of given function is drawn assuming elements of domain along x-axis. If a horizontal line (parallel, to x-axis) drawn anywhere in graph, intersects the graph at only one point then the function is oneone, otherwise it is many-one., , Mathematics—XII: Term-1

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ULTIPLE CHOICE QUESTIONS, , Choose and write the correct option in the following questions., , i., , 10., , 11., , 13., , 14., , The relation R in the set A = {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 0), (4, 4), (1, 3), (3, 3), (3, 2, is, , (a) reflexive and symmetric but not transitive, , (b) reflexive and transitive but not symmetric, , (c) symmetric and transitive but not reflexive, , (d) an equivalence relation, , If A = {a, b, c, dj, then a relation R = {(a, b), (b, a), (a, a)} on A is, , (a) symmetric only (b) transitive only, , (ey vefledive und transifive (d) symmetric and transitive only, , For real numbers x and y, define xRy if and only if x - y+/2 is an irrational number. Then the, , relation R is [NCERT Exemplar], (a) reflexive (b) symmetric (c) transitive (d) none of these, Consider the non-empty set consisting of children in a family and a relation R defined as aRb, if a is brother of b. Then R is LNCERT Exemplar], (a) symmetric but not transitive (b) transitive but not symmetric, , (c) neither symmetric nor transitive (d) both symmetric and transitive, , The maximum number of equivalence relation on the set A ={1,2,3} are [NCERT Exemplar], (a) 1 (8) 2 (c) 3 (a) 5, , Let L denotes the set of all straight lines in a plane. Let a relation R be defined by /Rm if and, only if / is perpendicular to m V J, m € L. Then Ris INCERT Exemplar], (a) reflexive (b) symmetric (c) transitive (d) none of these, , Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and, symmetric but not transitive is, , (a) 1 (b) 2 (c) 3 (a) 4, Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is/are, (a) 1 (2) 2 () 3 (d) 4, , Let A and B be finite sets containing m and 1 elements respectively. The number of relations that, can be defined from A to B is, , (a) 2" (o 2" () mn (a0, Set A has 3 elements and the set B has 4 elements. Then the number of injective mapping that, can be defined from A to B is [NCERT Exemplar], (a) 144 (b) 12 (c) 24 (d) 64, , The function f: R > R defined by fix) = 2" + 2!*! is, , (a) One-one and onto (6) Many-one and onto, , (c) One-one and into (d) Many-one and into, , If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one, and onto mapping from A to Bis, , (a) 720 (bv) 120 (0 (d) none of these, Which of the following functions from Z into Z is bijection? [NCERT Exemplar], (a) fle) =2° (0 flx) =x +2 (©) fx) = 2x41 (a) fle) =37 #1, Let f: [2, <) R be the function defined by f(x) = x - 4x + 5, then the range of fis, , [NCERT Exemplar], (a) R () [1,«) (©) 14, %) (d) [5,«), , Relations and Functions | 7

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8, , 15., , 16., , 17., , 18., , 19., , 20., , Zi., , a5:, , 26., , 27., , Let f: R > R be defined by flx) = x7 + 1. Then, pre-images of 17 and -3, respectively, are, [NCERT Exemplar], , (a) 9, 14,4} (2) 13,-31, 6 (c) {4,-4L > (d) {4,-4f, {2,-2}, , Let the function f :R —R be defined by f(x) = 2x+sin x for x € R. Then fis, , (a) one-one but not onto (b) onto but not one-one, , (c) neither one-one nor onto (d) one-one and onto, , Let R be the relation in the set N given by R ={(a, b): a = b—2, b > 6} choose the correct answer., , (a) (2,4)ER (0) B,8)ER () © 8)ER (a) (8,7)ER, , Let f: R— R be defined as f(x) = x*. Choose the correct answer, , (a) fis one-one onto (b) fis many one onto, , (c) fis one-one but not onto (d) fis neither one-one nor onto., , Let f: R— R be defined as f(x) = 3x, Choose the correct answer., , (a) fis one-one onto. (b) fis many one onto., , (c) f is one-one but not onto (d) f is neither one-one nor onto., , Let f: R——R defined by, f(x) = 2x? + 2x? + 300x +5 sin x then f is, (a) one-one onto (d) one-one into (c) many one onto (d) many one into, Let f:R— R be defined by f(x) = x" + 1. Then, pre-image of 5 and - 5, respectively are, (a) 6, {-2} 2) {G,-3h4 (©) {-2,2h6 (a) (1, -1}, {2,-2}, The domain of the function f : R—R defined by f(x) = vx?-4 is, (a) [-2,2] (2) (-2,2) (c) (2,2) U[2, 20) (d) (- 20, 00), , 3x, ifx>3, , Let f:R—R be defined by f(x) =)2", if1<x<3, x lest, Then f(— 2) + f(0) + f(2) + f(5) is equal to, (a) 0 (b) 17 () -4 (a) none of these, Let R is reflexive relation on a finite set A having element, and let there be m ordered pairs in, R. Then, , (a) m>n (b) msn (c) m=n (d) none of these, , The domain of the function f(x) = log, , (x7 - 1) is, , (a) (-3,-1)U (1,09) (b) [-3, -1) U[1, 29), , () (-3,-2)UC2,-1)UG, «) (4) [-3,-2)UG2,-1)U[L, 9), , Let f:R—[0, ba defined by fix) = tan” (x? +x + a), then the set of values of a for which f is, onto is 1, , (a) 0, x) ® Epo) (© 21 (a) none of these, , If the function f: R- R and g:R-— R are defined as, , i< 0, x rational, f= x, x irrational, , 0, x €irrational, ads t x €rational, then (f-g) is, (a) one-one onto. (b) many-one onto, (c) one-one but not onto (d) neither one-one nor onto., , Mathematics—XII: Term-1

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29,, , 31, , 32., , 34., , 35., , 36., , 37., , 38., , 39., , 41., , If a relation R on the set {1, 2, 3, 4} is defined by R = {(1, 2), (3, 4)}. Then R is, (a) reflexive (0) transitive (c) symmetric (d) none of these, , If the set A contains 4 elements and the set B contains 5 elements, then the number of one-one, and onto mappings from A to B is, , (a) 0 (0) 4 (c) 5* (d) none of these, Let A = {x, y, z) and B = {a, b} then the number of onto function from A to B is, (a) 0 (i) 3 () 6 (a) 8, , If A and B have 4 and 6 elements respectively then the number of one-one function from A to, Bis, , (a) 4° (») 64 (©) 360 (a) 240, , If A and B have 4 elements each then the number of one-one onto (bijective) function from A, to Bis, , (a) 0 (b) 24 () # (d) None of these, , If R is an equivalence relation on A, then R™ on A is, , (a) Transitive only —_(b) Symmetric only (c) Reflexive only (d) Equivalence relation, The relation “greater than” denoted by > in the set of integers is, , (a) Symmetric (b) Reflexive (c) Transitive (a) None of these, If R, and R, are symmetric relations in a set A, then R, UR, is, , (a) Reflexive (0) Symmetric (c) Transitive (d) None of these, The function {:R— R defined by f(x) = 4° + 4!"! is, , (a) one-one and into (b) one-one and onto, , (c) many one and into (d) many one and onto, , Identity relation R on a set A is, , (a) Reflexive only (b) Symmetric only (c) Transitive only — (a) Equivalence, , The relation “congruence modulo m” on the set Z of all integers is a relation of type, , (a) Reflexive only (b) Symmetric only (c) Transitive only — (d) Equivalence, , Let f: R—— | 0, es) defined by f(x) = tan™ (x* +x + 2a) then the set of values of ‘a’ for which, , fis onto, is, , , , , , 1 1 1, @ (-F) ) [-4,) © |-Z<) « [~), If the function f(x) satisfying (f(x))? - 4f()f’ (x) + (f(x)? = 0 then flx) equals, (a) ree tvee O) del? v5) ©) el2tvPe (a) nred- v3)", , x, , Let f :(-1,1)——~ B where f(x) = tan(> Z ) is one-one and onto, then B equals, , x aT an qe, © |o5| obEF OCF) — @ (05), The function y = — ay xER yeR is, , “ 14+lyl ,, , (a) One-one onto (b) Onto but net one-one, (c) One-one but not onto (d) None of these, A relation R in the set of non-zero complexs number is defined by z,Rz,@ = is real,, then R is, (a) Reflexive (b) Symmetric (c) Transitive (d) Equivalence, Number of onto (subjective) functions from A to B if (A) = 6 and (B) = 3 are, (a) 2°-2 (b) 3°-3 (c) 340 (d) None of these, , Relations and Functions | 9