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1 Sets, Relations and Functions

(i) Listing method In this method, the set is Sets represented by writing elements in a bracket and separatedbycomma(,). In our mathematical language, everything in this = whetherlivingornon-livingiscalledanobject. e.g., A SetofvowelsofEnglishalphabet Any collection of well defined objects is called a set. By = {,,,,}.a e i o u ‘well defined objects’, we mean that given a set and an This method is also known as Tabular method or object, it must be possible to decide whether or not the Rostermethod. object belongs to the set. The objects in set are called its (ii) Set builder method In this method, instead membersorelements. of listing all elements of a set, we write the set by some Sets are denoted by capital letters A, B, C etc., while special property or properties satisfied by all elements and the elements are denoted in general by small letters a, b, c, writeitas, A= {:()} x P x etc. where P() x isapropertywhichissatisfiedby x. Followingcollectionsaresets e.g., A = {,,,}1 2 3 4 ={:x x ∈ N and x < 5} (i) Thecollectionofallpositiveintegers. This method is also known as Rule method or Property (ii) ThecollectionofallcapitalsofstatesofIndia. method. Let A be any set of objects and let ‘a’ be a member of A, % In writing the elements of any set there is no consideration of then we write a∈ A and read it as ‘a belong to A’ or ‘a is an , e.g.,{,,,,}a e i o u and {,,,,}e i o u a aretwosamesets. element of A’ or ‘a is member of A’. If a is not an object of A, then we write a∉ Aand read as ‘a does not belong to A’ or ‘a isnotanelement of A’or‘a isnotmemberof A’. TypesofSets Somestandardnotationsforsomespecialpoints. (i) A set consisting of no element is (i) The set of all natural numbers i.e., the set of all called an empty set or null set or void set and is denoted by φ positiveintegersisdenotedby N. symbol or { }. = ∈ < < } = φ (ii) Thesetofallintegersisdenotedby Z or I. e.g., A{ x : x N and 3 x 4 . (iii) Thesetofallrationalnumbersisdenotedby Q. Asetwhichisnotemptyiscallednon-emptysetor non-voidset. (iv) Thesetofallrealnumbersisdenotedby R. + (ii) set A set consisting of only one (v) The set of all positive real numbers is denoted by R . elementiscalledasingletonset. (vi) Thesetofallcomplexnumbersisdenotedby C. e.g., {2},{0},{φ} (vii) The set of all positive rational numbers is denoted by Q+. % Theset{0}isnotanemptysetasitcontainsoneelement0. % Theset{φ}isnotanemptysetasitcontainsoneelement φ.

RepresentationofSets (iii) A set having finite number of Therearetwomethodstorepresentsets elementsiscalledfiniteset. e.g., A = {1,2,3}isafiniteset. (i)Listingmethod(ii)Setbuildermethod 2 NDA/NA Mathematics

(iv) Infinite set A set which is not finite is called % Whenever, we have to show that two sets A and B are equal i.e., = ⇔ ⊆ ⊆ aninfiniteset. A B ABBAand . = e.g., A Setofpointslieinaplaneisaninfiniteset. (x) Universal set In any discussion in , (v) Cardinal number of a finite set The we need a set such that all sets under consideration in that number of elements of a finite set A is called its cardinal discussion are its subsets. Such a set is called the universal numberanditisdenotedby n(A)or o(A). set and is denoted by U. (vi) Equivalent sets Two finite sets A and B are (xi) Power set The set of all the subsets of a given said to be equivalent if they have the same cardinal set A is said to be the power set of A and is denoted by P().A number.Thus,sets A and B areequivalentif n()()AB= n . e.g., If A = {,,}1 2 3 ,then (vii) Subset and super set The set B is said to PA(){,{},{},{},{,},{,},= φ 1 2 3 12 23{,},{,,}}3 1 1 2 3 . be subset of set A, if every element of set B is also an % Elementsofpowersetarethesubsetof A. ⊆ element of set A. Symbolically we write it as, BA or % Thepowersetofeachgivensetisalwaysnon-empty. AB⊇ ,where A issupersetof B. % If A is a finite set of n elements, then number of elements in PA() n (a) BA⊆ is read as B is contained in A or B is subset willbe 2 . of A or A issupersetof B. (b) AB⊇ is read as A contains B or B is a subset of A. VennDiagram Evidently, if A and B are two sets such that To express the relationship among sets in a ∈ ⇒ ∈ ⇒ x B x A, then B is subset of A. The symbol ‘ ’ stands perspective way, we represent them pictorially by means of for ‘implies’, we read it as x belongs to B implies that x diagrams,knownasVenndiagrams. belongsto A. e.g.,Let A = {,,,};1 2 3 4 B = {,,}1 2 4 U Here, B isasubsetof A. (viii) Proper subset The set B is said to be a proper subset of set A, if every element of set B is an element of A whereas every element of A is not an element The universal set is usually represented by a of B. rectangular region and its subsets by circle or closed We write it as BA⊂ and read it as ‘B is a proper subset boundedregionsinsidethisrectangularregion. of A’. Thus, B is a proper subset of A, if every element of B is an element of A and there is atleast one element in A which OperationsonSets is not in B. (i) of sets Let A and B are two sets, then Observe that A⊆ A i.. e, every set is a subset of itself, union of A and B is denoted by AB∪ and it consists of each butnotapropersubset. oneofwhichiseitherin A orin B orinboth A and B. e.g., Let AB={,,};{,},1 2 3 = 1 2 then BA⊂ . U % Nullsetisasubsetofeverysetandeachsetissubsetofitself. % Numberofsubsetofafinitesetof n elementsis 2n. A B (ix) Equal sets Two sets A and B are said to be equal, if each element of A is an element of B and each Thus, A∪ B ={ x:} x ∈ Aor x ∈ B elementof B isanelementof A. Clearly, x∈ A ∪ B ⇔ x ∈ Aor x ∈ B Thus, two sets A and B are equal, if they have exactly and x∉ A ∪ B ⇔ x ∉ Aand x ∉ B the same elements but the order in which the elements in Inthefigure,theshadedpartrepresents AB∪ . thetwosetshavebeenwrittendownisimmaterial. Itisevidentthat AABBAB⊆ ∪,. ⊆ ∪ Thus,if x∈ A ⇒ x ∈ B and y∈ B ⇒ y ∈ A, Example 1. If A = {,,}1 2 3 and B = {,,,}1 3 5 7 , then the ∪ then A and B areequal valueof AB is (a){4,5,7} (b){1,2,3,5,7} e.g., {4,8,10}={8,4,10} (c){1,2,3,5} (d){6,3,5,7} [The order in which the elements of a set is also Solution (b) Q AB={}{}1, 2 , 3 and = 1 , 3 , 5 , 7 immaterial] ∴ AB∪ = {}1,,,, 2 3 5 7 Sets,RelationsandFunctions 3

(ii) Intersection of sets The intersection of two (v) Symmetric difference of two sets The sets A and B, denoted by AB∩ is the set of all elements, symmetric difference of two sets A and B, denoted by AB∆ common to both A and B. is the set ()().ABBA− ∪ − U U A–B B–A

AB A B ∩ = ∈ ∈ Thus, A B{ x: x A and x B} Thus, ABABBA∆ =()() − ∪ − ={:x x ∉ A ∩ B} ∈ ∩ ⇔ ∈ ∈ Clearly, x A B x A and x B Theshadedpartrepresents AB∆ . and x∉ A ∩ B ⇔ x ∉ A or x∉ B = = Inthefigure,theshadedpartrepresents AB∩ . Example 3. If A {,,,,}1 3 5 7 9 and B {,,,,}2 3 5 7 11 , Itisevidentthat ABAABB∩ ⊆,. ∩ ⊆ thenfindthevalueof AB∆ is (a){5,7,11} (b){1,2,9,11} Example 2. If A = {,,,}1 2 3 4 and B = {,,}2 4 6 , then the (c){3,5,7} (d){1,3,5,11} ∩ valueof AB is Solution (b) Q A = {}1,,,, 3 5 7 9 and B ={,,,,}2 3 5 7 11 (a){2,4} (b){6,5} (c){2,3,4} (d){1,2,3} ∴ AB− ={ , }19 and BA− ={,}2 11 ∴ ABABBA∆ =()() − ∪ − Solution (a)Q A = {,,,}1 2 3 4 and B ={,,}2 4 6 ={,}{,}1 9 ∪ 2 11 ={,,,}1 2 9 11 ∴ AB∩ = {,}2 4 (iii) Disjoint sets Two sets U ComplementofaSet A and B are said to be disjoint sets, if they have no common element A B If U is a universal set and AU⊂ ,then complement set ′ − i.e., AB∩ = φ. of A isdenotedby AUAor . The disjoint sets can be U representedbyVenndiagramasshowninthefigure A B e.g.,let A = {,,}1 2 3 and B = {,}4 6 Here, A and B aredisjointsetsbecause AB∩ = φ . (iv) Difference of sets If A and B are two sets, Thus, A′ = U − A ={ x:,} x ∈ U but x ∉ A then their difference AB− is the set of all those elements Itisclearthat x∈ A ′ ⇔ x ∉ A of A whichdonotbelongto B. % φ =U ′ % φ′ = U % ()AA′ ′ = U % AAU∪ ′ = % AA∩ ′ = φ A–B LawsofAlgebraofSets AB If A,B and C arethreesets,then Thus, A− B ={:} x x ∈ Aand x ∉ B 1. Idempotent laws ∈ − ⇔ ∈ ∉ Clearly, x A B x Aand x B. (a) AAA∪ = (b) AAA∩ = − Inthefigure,theshadedpartrepresents AB. 2. Identitylaws Similarly, the difference BA− is the set of all those (a) AA∪ φ = (b) AUA∩ = elementsof B thatdonotbelongto A i.e., 3. Commutativelaws B− A ={:} x x ∈ Band x ∉ A (a) ABBA∪ = ∪ (b) ABBA∩ = ∩ U 4. Associativelaws B–A (a) ()()ABCABC∪ ∪ = ∪ ∪ (b) ABCABC∩()() ∩ = ∩ ∩ B A 5. Distributivelaws Inthefigure,theshadedpartrepresents BA− . (a) ABCABAC∪()()() ∩ = ∪ ∩ ∪ ∩ ∪ = ∩ ∪ ∩ e.g., if A = {,,,,}1 3 5 7 9 and B = {,,,,},2 3 5 7 11 then (b) ABCABAC()()() A −B = {,}1 9 and BA− = {,}.2 11 6. De-Morgan’slaws (a) ()ABAB∪ ′ = ′ ∩ ′ (b) ()ABAB∩ ′ = ′ ∪ ′ 4 NDA/NA Mathematics

7. (a) ABAB− = ∩ ′ (b) BABA− = ∩ ′ Facts Related to Cartesian Product (c) ABAAB− = ⇔ ∩ = φ (d) ()ABBAB− ∪ = ∪ (e) ()ABB− ∩ = φ If A, B and C are non-empty sets, then (f) ()()()()ABBAABAB− ∪ − = ∪ − ∩ 1. (a) ABCABAC×()()() ∪ = × ∪ × 8. (a) ABCABAC−()()() ∪ = − ∪ − (b) ABCABAC×()()() ∩ = × ∩ × (b) ABCABAC−()()() ∩ = − ∩ − 2. ABCABAC×()()() − = × − × (c) ABCABAC∩()()() − = ∩ − ∩ 3. ABBAAB× = × ⇔ = (d) ABCABAC∩()()()∆ = ∩ ∆ ∩ 4. If ABABABBA⊆ ⇒ × ⊆()() × ∩ × Important Results 5. If ABACBC⊆ ⇒ × ⊆ × 6. AB⊆ and CDACBD⊆ ⇒ × ⊆ × If A, B and C are any three finite sets, then 7. ()()()()ABCDACBD× ∩ × = ∩ × ∩ 1. n()()()() A∪ B = n A + n B − n A ∩ B 8. ABCABAC×()()() ′ ∪ ′ ′ = × ∩ × 2. n()()(), A∪ B = n A + n B if and only if AB∩ = φ 9. ABCABAC×()()() ′ ∩ ′ ′ = × ∪ × 3. n()()() A− B = n A − n A ∩ B 10. If n elements are common in A and B, then in AB× and ∆ = − + − = + − ∩ 4. n()()() A B n A B n B A n()()() A n B2 n A B BA× , n2 elements will be common. 5. nABC()()()()()∪ ∪ = nA + nB + nC − nAB ∩ −n()()() B ∩ C − n A ∩ C + n A ∩ B ∩ C Relation 6. n()()() A′ ∪ B ′ = n U − n A ∩ B Let A and B are two non-empty sets, then a relation R 7. n A′ ∩ B ′ = n U − n A ∪ B ()()() from A to B isasubsetof AB× . Equivalently, any subset of AB× is relation from A to B. Example 4. If in a group of 850 persons, 600 can speak Thus, R isarelationfrom A to B ⇔RAB ⊆ × Hindi and 340 can speak Tamil. Then, the number of persons ⇔ ⊆ ∈ ∈ canspeakbothHindiandTamilis R{(,):,} a b a A b B (a)40 (b)90 (c)85 (d)120 If (,),a b∈ R then we write a R/ b which is read as ‘a is related to b by the relation R’. If (,),a b∉ R then we write ab Solution (b) Let A and B denote the sets of persons who can andwesaythat a isnotrelatedto b bytherelation R. speakHindiandTamil,respectively. Then, n(); A =600 n() B =340 and n() A∪ B =850 DomainandRangeofRelation If R is a relation from set A to set B, then the set of all A–B B–A first coordinates of elements of R is called the domain of R, while the set of all second coordinates of elements of R is B A AB∩ calledtherangeof R, ∴ Domain (){:(,)}R= a ∈ A a b ∈ Rfor some b ∈ B n()()()() A∩ B = n A + n B − n A ∪ B and Range (){:(,)}R= b ∈ B a b ∈ Rfor some a ∈ A =600 + 340 − 850 = 90 Thus,90personscanspeakbothHindiandTamil. Codomain of a relation If R be a relation from A to B,then B iscalledthe codomain ofrelation R. OrderedPair TypesofRelation Two elements a and b listed in a particular order, is called an ordered pair and it is denoted by (,)a b . In an (i) Empty relation Since, φ ⊂AA × , it follows ordered pair (,);a b ‘a’ is regarded as the first element and that φis a relation on A, called the empty or void relation. ‘b’ thesecondelement. e.g.,Let A = {1,2}, B = {1,3} Itisevidentfromthedefinitionthat Let R = {(1,1),(1,3),(2,1),(2,3)} (i) (,)(,)a b≠ b a ⇔ a ≠ b Here, R = A × B = ⇔ = = (ii) (,)(,)a b c d a c, b d Hence, R istheuniversalrelationfrom A to B. CartesianProduct (ii) Universal relation Since, AAAA× ⊆ × , it follows that AA× is a relation on A, is called the universal Let A and B be two non-empty sets. The cartesian relation. product of A and B, denoted by AB× is defined as the set of (iii) Identity relation The relation I= {(,); a a allorderedpairs (,)a b ,where a∈ A and b∈ B. A a∈ A} is called the identity relation to A. Symbolically, A× B ={(,); a b a ∈ A and b∈ B} e.g., If A = {,,}1 2 3 , then the identity relation on A is Thus,(a, b) ∈A × B ⇔ a ∈ Aand b ∈ B = givenby IA {(,),(,),(,)}.1 1 2 2 3 3 Sets,RelationsandFunctions 5

(iv) Inverse relation If R is a relation on A, then Transitivity fails only when there exists a, b and csuch the relation R−1 on A, defined by R−1 ={(,):(,)} b a a b ∈ R is that aRb, bRc but a R/. c calledaninverserelationto A. e.g., let A = {,,}1 2 3 andtherelation − Clearly,domain ()R 1 = range(R) R = {(,),(,),(,)}.1 2 2 1 1 1 and range ()R−1 = domain(R) Then, R isnottransitive,since R, (,),(,)2 1∈RR 1 2 ∈ but (,).2 2 ∉R Example 5. Let A = {,,}1 2 3 and let R = {(1 , 2 ),( 2 , 2 ), (3 , 1 ), − (3 , 2 )},thenthedomainandrangeof R 1 is EquivalenceRelation (a){2,1},{1,2,3} (b){1,2},{1,2,3} A relation R on a set A is said to be an equivalence (c){1,2,3}{2,3} (d){2,3}{1,2} relation,if Solution (a) Q A ={,,}1 2 3 and R ={(,1 2 ),( 2 , 2 ),( 3 , 1 ), (3 , 2 )}. (i) R isreflexive i.e., (,),a a∈ R ∀ a ∈ A Then, R beingasubsetof AA× , itisarelationon A. (ii) R is symmetric i.e.,(,)(,),,a b∈ R ⇒ b a ∈ R ∀ a b ∈ A Clearly,1RRR 2;; 2 2 3 1and 3R 2. (iii) R istransitive i.e., (,),(,)(,)a b b c∈ R ⇒ a c ∈ R ∴ = = Domain (){,,}R 1 2 3 andrange (){,}R 2 1 Example 6. −1 = The relation ‘>’ on the set R of all real Also, R {(2 , 1 ),( 2 , 2 ),(, 1 3 ),( 2 , 3 )} numbersis − − Domain (){,}R 1 = 2 1 andrange (){,,}R 1 = 1 2 3 (a)transitive (b)reflexive (c)symmetry VariousTypesofRelations (d)anti-symmetric (i) Reflexive relations A relation R on a set A is > ∈ ∀ ∈ Solution (a) The relation ‘>’ on R is transitive, since a b and said to be a reflexive relation, if (,),.a a R a R It should > ⇒ > be noted, if ∃ any a∈ A such that (,),a a∉ R then R is not b c a c. reflexive. Since,norealnumbercanbegreaterthanitself,therelation isnotreflexive. e.g., let A = {,,}1 2 3 and R = {(,),(,)}1 1 2 2 Also, a> b⇒/, b> a Then, R isnotreflexive,since 3∈ A but (,)3 3 ∉R. e.g., 3> 2 but 2>/. 3 (ii) Symmetric relations A relation R on a set So,itisnotsymmetric. A is said to be a symmetric relation on A, if (,)(,),,.x y∈ R ⇒ y x ∈ R ∀ x y ∈ R i.e., if R related to y, then y is also R related to x. It should be noted that R is CompositionofRelations −1 symmetric, iff RR= . If R and S are relations from A to B and B to Let A = {1,2,3} C respectively, then SoR is a relation from A to C, which is definedasfollows Let R = {(,),(,)}1 2 2 1 1 (,)a c∈ SoR ⇔ ∃ b ∈ B s.t. (,)a b∈ R and (,).b c∈ S R = {(,),(,),(,),(,)}1 2 2 1 1 3 3 1 2 Thisrelationisknownascompositionof R and S. R Here, 1 and R2 aresymmetricrelationson A. (iii) Anti-symmetric relations A relation R on a set A is said to be an anti-symmetric relation, if (,)a b∈ R Function and(,).b a∈ R ⇒ a = b Thus, if a≠ b,then a may be related to Let A and B are two non-empty sets, then subset of b or b mayberelatedto a, butneverboth. A× B i.., e f is known as a function from A to B iff ∀a ∈ A ∃ e.g., let N be the set of natural numbers. A relation auniqueelementin B suchthat (,).a b∈ f RNN⊆ × isdefinedby xRy iff x divides y(..,/) i e x y Function is also known as mapping, transformations, Then, xR y, yRx⇒ x divides y, y divides x. operators. → f → ⇒ x= y Functionisdenotedby f: A B or AB. e.g.,let A = {1,2,3} = Let R1 {(1,2),(1,3),(1,1)} Domain, Codomain andRangeof = R2 {(1,2)} aFunction Here, R1 and R2 areanti-symmetricrelationson A. Let f:. A→ B Then, the set A is known as the domain of (iv) Transitive relations A relation R on a set A f and the set B is known as the codomain of f. The set of all is said to be a transitive relation, if (,),a b∈ R f-images of elements of A is known as the of f or (,)(,)b c∈ R ⇒ a c ∈ R. imagesetof A under f andisdenotedby f(). A In other words, if a is related to b, b is related to c, then Thus, f(){():} A= f x x ∈ A = Rangeof f. a isrelatedto c. Clearly, f(). A⊆ B 6 NDA/NA Mathematics

VariousTypesofFunctions (b) it is onto i.e., ∀y ∈ B,there exists y∈ A such that f(). x= y (i) Many-one function Let f:. A→ B If two or (vi) Constant function Let f: A→ B is defined more than two elements have the same image in B, then f is in such a way that all the elements in A have the same saidtobemany-onefunction. imagein B,then f issaidtobeaconstantfunction. e.g., the function f: A→ B given by f() x= x2 is a Thus, f() x= c for every x∈ A, where c is a fixed number many-onefunction. Clearly,domainof f= A andrangeof f= {} c . A B f Thus, a function f: A→ B is a constant function, if 1 1 rangeof f isasingletonset. –1 = = → 2 4 e.g., A {1, 2, 3} and B {5, 7}, let f: A B defined by 7 f(), x=5 ∀ x ∈ A. Then, all the elements in A have the same image in B. (ii) One-one function (injective) Let f:. A→ B So, f isaconstantfunction. Then, f is said to be one-one function or an injective, if (vii) Identity function Let A be a non-empty → = ∀ different elements of A have different images in B. set. Then, the function, defined by IAA:.() A A I x x, ∈ Thus, f: A→ B isone-one x A, iscalledanidentityfunctionon A. This is clearly a one-one onto function with domain A ⇔ a≠ b ⇒ f()() a ≠ f b , ∀a, b ∈ A andrange A. ⇔ f()() a= f b ⇒ a = b, ∀a, b ∈ A → = (viii) Equal functions Two functions f and g are e.g., the function f: A B given by f() x2 x is an said to be equal, written as f= g, if they have the same one-onefunction. domainandtheysatisfythecondition f()() x= g x , ∀ x. f 1 2 (ix) Even and odd functions A function –1 –2 f: A→ Bis said to be an even or odd function according as 3 6 − = ∀ ∈ − = − ∀ ∈ 7 f()(), x f x x A and f()(), x f x x A, respectively. (surjective) Let f:. A→ B If (iii) Onto function Example 7. Let N be the set of all natural numbers. If every element in B has atleast one preimage in A, then f is f:() N→ Ndefined as f x = 2 x, ∀ x∈ N, then f is saidtobeanontofunction. (a)one-oneonto (b)many-oneonto Thus, f: A→ Bis a surjective, iff for each b∈ B, ∃ a ∈ A (c)one-oneinto (d)many-oneinto suchthat f() a= b clearly, f isonto ⇔ range ().f= B Solution (a)Clearly,each x∈ N hasitsuniqueimage 2x∈ N. f 1 3 So, f isonto. –1 –3 Also, f isone-one,since 2 6 = ⇒ = ⇒ = f()() x1 f x 22 x 1 2 x 2 x 1 x 2 Hence, f isone-oneandonto. e.g., the function f: A→ B is given by f() x= 3 x is an ontofunction. (x) Inverse function Let f be a one-one onto functionfrom A to B. (iv) Into function Let f:. A→ B If there exists f even a single element in B having no preimage in A, then f x y issaidtobeanintofunction. 2 –1 e.g., the function f: A→ B given by f() x= x is an into x = f() y −1 y= f ( x ) f function. AB A B f Let y be an arbitrary element of B. Then, f being onto, 1 1 ∈ = –1 thereexistsanelement x A,suchthat f(). x y 2 4 Also, f beingone-one,this x mustbeunique. 7 Thus, for each y∈ B, there exists a unique element x∈ A,suchthat f(). x= y (v) Bijective function A one-one and onto So,wemaydefineafunction,denotedby f − 1 as functionissaidtobe bijective. −1→ − 1 = ⇔ = A bijective function is also known as a one-to-one f:,()(). B Asuch that f y x f x y correspondence. Theabovefunction f − 1 iscalledtheinverseof f. Inotherwords,afunction f: A→ B isa bijection,if (a) itisone-one i.e., f()() x= f y ⇒ x = y, ∀x,. y ∈ A % Afunction f isinvertibleifandonlyif f isone-oneonto. Sets,RelationsandFunctions 7

Example 8. The inverse of the function f: R→ R defined by CompositionofFunctions = − f() x4 x 7 is Let A, B and C be three non-empty sets. Let f: A→ B x + 4 x + 7 → → ∈ (a) (b) and g:. B C Since, f:, A B for each x A,there exists a 2 4 unique element g[()] f x of C. Thus, for each x∈ A, there is x − 4 associated a unique element g[()] f x of C. Thus, from f and (c) (d)doesnotexist 5 g, we can define a new function from A to C. This function is called the product or composite of f and g, denoted by gof Solution (b) Wehave, f() x=4 x − 7, x∈ R anddefinedby isone-one ∈ → = ∈ f Let x1, x 2 R ():()()[()]gof A Csuch that gof x g f x forall x A. = A f B g C Now, f()() x1 f x 2 ⇒ − = − 4x1 7 4 x 2 7 x f() x g( f ( x )) ⇒ = ⇒ = 4x1 4 x 2 x 1 x 2 ∴f isone-one. gof f isonto Let y ∈codomain R Let f() x= y ⇒4 x − 7 = y PropertiesofCompositionofFunctions y + 7 1. The product of any function with the identity ⇒ x = ∈R 4 functionisthefunctionitself. ∴ 2. The product of any invertible function f with its f isonto. − − 1 Hence, f isinvertible i.e., f 1 exists. inversefunction f isanidentityfunction. − Tofind f1 :() f x= y ⇒4 x − 7 = y 3. Compositeoffunctionsisassociative. → → y + 7 4. Let f: A B and g : BA, such that gof is an ⇒ x = identity function on A and fog is an identity function 4 − on B.Then, g= f 1. − y + 7 − ⇒ f1()[()()] y = Q f x=4 ⇔ x = f1 y → → 4 5. Let f: A B and g: B C be a one-one onto functions. − x + 7 ⇒ f1() x = −1= − 1 − 1 4 Then, gof isalsoone-oneontoand ()gof f og

Comprehensive Approach

I A natural number p is a prime number, if ‘p’ is greater than one and I For two relations R and S, the composite relation RoS, SoR may be itsfactorsareoneand ‘p’ only. voidrelation. I Finite sets are equivalent sets only when they have equal numbers I Theproductoftwoevenoroddfunctionisanevenfunction. ofelements. I Theproductofanevenandanoddfunctionisanoddfunction. I Equal sets are equivalent sets, but equivalent sets may not be equal I Every function f(x) can be expressed as the sum of an even and an sets. oddfunction. n I Number of proper subsets of a set containing ‘n’ elements is2− 1. I If A and B have n and m distinct elements respectively, then the n I If AB⊆ , we may haveBA⊆ but, if AB⊂ ,we cannot haveBA⊂ . numberofmappingsfrom A to B isequalto m . I {,}x y and {,}y x are equals sets but (,)x y and (,)y x are not equal I If A and B have n equal number of distinct elements, then the orderedpair. numberofmappingsfrom A to B isequalto nn. I If ‘A’ isanyset,then AA⊆ istruebut AA⊂ isfalse. I If A and B have n equal number of distinct elements, then the number n()() A⋅ n B I Totalnumberofrelationsfromset A toset B isequalto 2 . of objective functions from A to B is equal to n !. I The universal relation on a non-empty set is always reflexive, I The number of one-one functions that can be defined from a finite set n() B ≥ symmetricandtransitive. A into a finite set B is Pn() A , if n()() B n A and zero, otherwise. I The identity relation on a non-empty set is always reflexive, I The number of onto functions that can be defined from a finite set symmetricandtransitive. A containing n elements on finite set B containing 2 elements =n − I The identity relation on a non-empty set is always anti-symmetric. 2 2. I I If R is relation from A to B and S is a relation from B to C, then If two curves do not intersect each other, then intersection of two − − − ()RoS 1 = S1 oR 1. setsisanullset. Exercise

LevelI 1. If AB× = {(,),(,),(,),11 12 13(,),(,2 1 2 2 ),(, 2 3 )}, then A not taken Science or Mathematics or both Science isequalto andMathematics,isequalto (a){1,2} (b){1,2,3} (a)90 (b)10 (c){2,3} (d)Noneofthese (c)30 (d)20 2. If A = {}1,, 2 3 and B = {}3 4,,then()()ABAB∪ × ∩ is 13. Therelation R onaset A = {,,,}1 2 3 4 isdefinedas (a){3,3} {(1,1), (,),13 (2 , 2 ),( 2 , 3 ),( 3 , 1 )( 3 , 2 )}. Then, R is (b){(1,3),(2,3),(3,3),(1,4),(2,4),(3,4)} (a)reflexive (b)symmetric (c){(1,3),(2,3),(3,3)} (c)anti-symmetric (d)transitive (d){(1,3),(2,3),(3,3),(4,3)} 14. If A, B and C are non-empty sets such that AC∩ = φ, 3. Whichofthefollowingiscorrect? thenwhatis ()()ABCB× ∩ × equalto? (a) ABAB∩ ⊂ ∪ (b) ABAB∩ ⊆ ∪ (c) ABAB∪ ⊂ ∩ (d)Noneofthese (a) AC× (b) AB× (c) BC× (d) φ 4. If n(),(), A=8 n A ∩ B = 2 then n() A− B isequalto (a)8 (b)2 15. If A={4 n + 2 | n is a natural number} and B= { n |3 n (c)6 (d)9 isanaturalnumber},thenwhatis ()AB∩ equalto? 2 + 5. A set contains n elements. Then, the power set (a) {12n 6 n | n isanaturalnumber} contains (b) {24n− 12 | n isanaturalnumber} (a) n 2 elements (b) n elements (c) {12n2 − 6 n | n isanaturalnumber} n n (c) ()2− 1 elements (d) 2 elements (d) {12n− 6 | n isanaturalnumber} 6. If A = {,,,}1 2 3 4 and B = {,,}5 6 7 , then number of 16. If A and B are two disjoint sets, then which one of the relationsfrom A to B isequalto followingiscorrect? 4 3 7 12 (a) 2 (b) 2 (c) 2 (d) 2 (a) ABAAB− = −() ∩ (b) BABA− ′ = ∩ 7. If φ is a null set, then which one of the following is (c) ABABB∩ =() − ∩ (d)Allofthese correct? 17. If the of a set A is 4 and that of a set B is 3, (a) φ = 0 (b) φ = { }0 thenwhatisthecardinalityoftheset AB∆ ? (c) φ ={} φ (d) φ = {} (a)1 8. If A = {a, b, c}, then what is the number of proper (b)5 subsetsof A? (c)7 (a)5 (b)6 (d) Cannot be determined as the sets A and B are not (c)7 (d)8 given. 9. If A = {,,,}1 2 5 6 and B = {,,}1 2 3 , then what is 18. If A and B are finite sets, then which one of the ()()ABBA× ∩ × equalto? followingisthecorrectequation? − = − (a){(1,1),(2,1),(6,1),(3,2)} (a) n()()() A B n A n B (b) n()() A− B = n B − A (b){(1,1),(1,2),(2,1),(2,2)} − = − ∩ (c){(1,1),(2,2)} (c) n()()() A B n A n A B (d) n()()() A− B = n B − n A ∩ B (d){(1,1),(1,2),(2,5)(2,6)} [()n A denotesthenumberofelementsin A] 10. If E is the universal set and ABC= ∪ , then the set ((((())))EEEEEA− − − − − isthesameastheset 19. Whichoneofthefollowingiscorrect? (a) BCc∪ c (b) BC∪ The relation R = {(,),(,),(,)}1 1 2 2 3 3 on a set = (c) BCc∩ c (d) BC∩ A {,,}1 2 3 is (a)onlyreflexive 11. If a set A contains 3 elements and another set B (b)onlysymmetrics contains 6 elements, then the number of elements in (c)onlytransitive AB∪ would be (d)reflexive,symmetricandtransitive (a)9 (b)either8or9 = = (c)either7or8or9 (d)either6or7or8or9 20. If X {multiplesof2}, Y {multiplesof5}, Z={multiplesof10},then XYZ∩() ∩ isequalto 12. In a class of 100 students, 70 have taken Science, 60 (a){multiplesof10} (b){multiplesof5} have taken Mathematics, 40 have taken both Science (c){multiplesof2} (d){multiplesof20} and Mathematics. The number of students who have Sets,RelationsandFunctions 9

x 21. Let X be any non-empty set containing n elements. 32. Thefunction f() x = from R to R is Then,whatisthenumberofrelationson X? x2 + 1 2 (a) 2n (b) 2n (c) 22n (d) n 2 (a)one-oneaswellasonto 22. Whichofthefollowingisanullset? (b)ontobutnotone-one (a) {x :| x |<1 , x ∈ N } (c)neitherone-onenoronto (b) {x :| x |=5 , x ∈ N } (d)one-onebutnotonto (c) {:,}x x2 =1 x ∈ Z 33. Let A ={,,} −1, 2 5 8 , B = {,,,}01, 3 6 7 and R be the (d) {:,}x x2 +2 x + 1 = 0 x ∈ R relation ‘is one less than’ from A to B, then how many elementswill R contain? 23. The order of a set A is 3 and that of a set B is 2. What (a)2 (b)3 (c)5 (d)9 isthenumberofrelationsfrom A to B? = = − = (a)4 (b)6 (c)32 (d)64 34. If n() A 115, n() B 326, n() A B 47, then what is n() A∪ B equalto? 24. Thesetofintelligentstudentsinaclassis (a)373 (b)165 (c)370 (d)394 (a)anullset (b)asingletonset 35. If PA() denotes the power set of A and A is the void (c)afiniteset set, then what is number of elements in (d)notawelldefinedcollection PPPPA{{{()}}}? 25. If AP= {}1, 2 , where P denotes the power set, then (a)0 (b)1 (c)4 (d)16 whichoneofthefollowingiscorrect? = ∈ ∩ 36. If Na { ax | x N },thenwhatis NN12 8 equalto? (a) {,}1 2 ⊂ A (b) 1 ∈ A (c) φ ∉ A (d) {,}1 2 ∈ A (a) N (b) N (c) N (d) N 26. If A and B are any two sets, then what is the value of 12 20 24 48 AAB∩() ∪ ? 37. ConsiderthefollowingVenndiagram (a)Complementof A (b)Complementof B E (c) B (d) A 27. Let A= {: x x is a digit in the number 3591} A B B={:,}, x x ∈ N x < 10 which of the following is not correct? (a) AB∩ = {,,,}1 3 5 9 (b) AB− = φ If|EA |=42 ,| | = 15 , |B |= 12 and|AB∪ | = 22 , then (c) B −A = {,,,,}2 4 6 7 8 the represented by the shaded portion in the (d) AB∪ = {,,,,}1 2 3 5 9 aboveVenndiagramis (a)25 (b)27 (c)32 (d)37 28. If A={(,):,} x y y = ex x ∈ R and B={(,):,} x y y = e− x x ∈ R ,then AB∩ is 38. If A, B and C are three sets and U is the universal set such that n() U = 700, n() A = 200 , n() B = 300 and (a)emptyset (b)singletonset n() A∩ B = 100,thenwhatisthevalueof ()AB′ ∩ ′ ? (c)notaset (d)Noneofthese 29. The relation R defined on set A={ x :| x | <3 , x ∈ Z } by (a)100 (b)200 (c)300 (d)400 R={( x , y ) : y = | x |} is = (a) {(,),(,),(,),(,),(,)}−2 2 − 1 1 0 0 1 1 2 2 39. Let A{: x x is a square of a natural number and x is (b) {(,),(,),(,),(,),(,)−2 − 2 − 2 2 − 1 1 − 1 − 1 0 0 , less than 100} and Bis a set of even natural numbers. Whatisthecardinalityof AB∩ ? (,),(,),(,),(,)}1− 2 1 2 2 − 1 2 − 2 (a)4 (b)5 (c) {(,),(,),(,)}0 0 1 1 2 2 (c)9 (d)Noneofthese (d)Noneoftheabove 40. If X={(4n − 3 n − 1 )| n ∈ N } and 30. Let A = {,,,}2 3 4 5 and R = {(,),(,),(,),(,)}2 2 3 3 4 4 5 5 bearelationin A.Then, R is Y={9 ( n − 1 )| n ∈ N },then XY∪ equalsto (a)reflexive (b)symmetric (c)transitive (d)Noneofthese (a) X (b) Y (c) N (d)anullset 31. For non-empty subsets A, B and C of a set X such that ABBC∪ = ∩ , which one of the following is the 41. Sets A and B have n elements in common. How many strongestinferencethatcanbederived? elementswill ()AB× and ()BA× haveincommon? (a) ABC= = (b) ABC⊆ = (c) ABC= ⊆ (d) ABC⊆ ⊆ (a)0 (b)1 (c) n (d) n 2 10 NDA/NA Mathematics

42. What is the number of proper subsets of a given finite 45. In a class containing 120 students, 65 students drink setwith n elements? tea and 84 students drink coffee. If x students drink (a) 2n − 1 (b) 2n − 2 bothteaandcoffee,thenwhatisthevalueof x? (c) 2n − 1 (d) 2n − 2 (a)39 (b)65 (c) 29≤x ≤ 65 (d) 29≤x ≤ 84 43. If A, B and C are three sets, such that ABAC∪ = ∪ and ABAC∩ = ∩ , then which one of the following is 46. If A and B are two sets satisfying ABBA− = − , then correct? whichoneofthefollowingiscorrect? (a) AB= only (b) BC= only (a) A = φ (b) AB∩ = φ (c) AC= only (d) ABC= = (c) AB= (d)Noneofthese 44. If A, B and C are three sets, then ABC−() − equals to 47. Two finite sets have m and n elements, respectively. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. (a) ABC−() ∩ (b) ()ABC− ∪ (c) ()()ABAC− ∪ ∩ (d) ()()ABAC− ∪ − Whatarethevaluesof m and n,respectively? (a)7,6 (b)6,3 (c)5,1 (d)8,7 LevelII 1. Consider the following in respect of subsets A and B 5. Whatistherangeof f( x )= cos2 x − sin 2 x ? of X (a)[2,4] (b) [,]− 1 1 I. ABABA⊆ ⇔ ∪ = (c) [,]− 2 2 (d) (,)− 2 2 II. ()ABAB∪c = c ∩ c = 6. Which one of the following is the union of the closed III. ABBA//  1 1 IV. ABABAB∪ = ∩ , iff = sets 2 +,,,,...?10 −  n = 1 2  n n Whichofthesearecorrect? (a)[2,10] (b)(2,10) (a)IandIII (b)IandIV (c)IIandIII(d)IIandIV (c)[2,10) (d)(2,10] 2. Considerthefollowingstatements 7. Q =(sinθ + cos θ ). Which one of the following is the I.Allpoets(P)arelearnedpeople(L). correctrangeof Q? II.Alllearnedpeople(L)arehappypeople(H). (a) −2 ≤Q < 2 (b) −2

(a) PL (b) P L

H P Q

R (c) PL (d) PL H

H (a) ()()PQPR∩ ∩ ∩ 3. If R be a relation on NN× defined by (,)(,)a b R c d if (b) (())(())PQRPRQ∩ − ∪ ∩ − ∪ − ∩ ∩ − andonlyif ad= bc; then R is (c) (())(())PQRPRQ (d) (())(())PQRPQR∩ ∪ ∩ ∪ − (a)anequivalencerelation (b)symmetricandtransitivebutnotreflexive 9. Consider the following with regard to a relation R on (c)reflexiveandtransitivebutnotsymmetric a set of real numbers defined by xRy if and only if + = (d)reflexiveandsymmetricbutnottransitive 3x 4 y 5 = = I. 0R 1 4. If A {,,,}1 2 3 4 and B {,,}2 3 5 , then identify the 1 correct relation, among the following from A to B II. 1R < 2 givenby xRy, ifandonlyif x y 2 3 = III. R (a) R {}(,),(,),(,),(,)1 2 1 3 2 2 2 3 3 4 (b) R = {}(,),(,),(,),(,)3 2 3 3 3 4 3 5 Whichoftheabovearecorrect? (c) R = {}(,),(,),(,),(,)1 2 1 3 2 3 2 5 (a)IandII (b)IandIII (d) R = {}(,),(,),(,),(,)1 3 1 5 3 2 4 2 (c)IIandIII (d)I,IIandIII Sets,RelationsandFunctions 11

10. Let M be the set of men and R is a relation ‘is son of’ 17. Considerthefunction f:{,} R → 0 1 suchthat definedon M.Then, R is 1, ifx is rational f() x =  (a)anequivalencerelation 0, ifx is irrational (b)asymmetricrelationonly (c)atransitiverelationonly Whichoneofthefollowingiscorrect? (d)Noneoftheabove (a)Thefunctionisone-oneinto 11. Let N denote the set of natural numbers and (b)Thefunctionismany-oneinto A={:} n2 n ∈ N and B={:} n/32 n ∈ N . Which one of (c)Thefunctionisone-oneonto (d)Thefunctionismany-oneonto thefollowingiscorrect? 18. Which one of the following functions f: R→ R is (a) ABN∪ = injective? (b)Thecomplementof ()AB∪ isaninfiniteset 2 (c) AB∩ mustbeafiniteset (a) f( x )= | x |, ∀x ∈ R (b) f() x= x , ∀x ∈ R (d) AB∩ mustbeapropersubsetof {:}m6 m∈ N (c) f(), x = 11 ∀x ∈ R (d) f(), x= − x ∀x ∈ R 12. The relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), 19. If A= {,,} a b c and R= {(,),(,),(,),(,), a a a b b c b b (1,3)}onaset A = {1,2,3}is (,),(,)}c c c a is a binary relation on A, then which one (a)reflexive,transitivebutnotsymmetric ofthefollowingiscorrect? (b)reflexive,symmetricbutnottransitive (a) R isreflexiveandsymmetricbutnottransitive (c)symmetric,transitivebutnotreflexive (b) R isreflexiveandtransitivebutnotsymmetric (d)reflexivebutneithersymmetricnortransitive (c) R is reflexive but neither symmetric nor 13. Let R be a set of real numbers and let S be a relation transitive definedon R asfollows (d) R isreflexivesymmetricandtransitive x Sy, ifandonlyif, x2+ y 2 = 1 20. If α,, β ξand η arenon-emptysets,then Whichoneofthefollowingstatementsiscorrect? (a) ()()()()α× β ∪ ξ × η = α × β ∩ ξ × η α× β ∩ ξ × η = α × ξ ∩ β × η (a) S isareflexiverelation (b) ()()()() α∩ β × ξ ∩ η = α × ξ ∪ β × η (b) S isasymmetricrelation (c) ()()()() α∩ β × ξ ∩ η = α × η ∩ β × ξ (c) S isatransitiverelation (d) ()()()() (d) S isananti-symmetricrelation 21. In a Euclidean plane, which one of the following is 14. If F() n denotes the set of all divisors of a natural notanequivalencerelation? number n, what is the least value of y satisfying (a) Parallelism of lines (a line being deemed parallel [()()]()]F20∩ F 16 ⊂ F y ? toitself) (a)2 (b)1 (c)4 (d)6 (b)Congruenceoftriangles (c)Similarityoftriangles 15. HE (d)Orthogonalityoflines T 22. Whichoneofthefollowingiscorrect? (a) The relation R0 defined on the set of real = 2+ 2 = numbers as R0 {(,) a b such that a b 1 for all a,} b∈ R isanequivalencerelation K (b) The relation R0 defined on the set of real = − ≤ 1 numbers as R0 {(,) a b such that|a b | for The Venn diagram shown above represents four sets 3 ∈ of people who can speak Telugu (T), English (E), all a,} b R isanequivalencerelation Hindi (H) and Kannada (K). What does the marked (c) The relation I0 defined on the set of integers as 2 − +2 = ∈ regionrepresent? I1 lo I 2: I 1 3 I1 I 2 2 I 2 0 for all III1, 2 is an (a) People,whocanspeakHindiandKannadaonly equivalencerelation (b) People, who can speak English, Telugu and (d) We define AEBc by the open sentence : A is Kannadaonly cardinallyequivalentto B onthefamilyofsets (c) People,whocanspeakHindiandEnglishonly Then, the relation Ec on family of sets is an (d) People, who can speak Hindi, English and equivalencerelation Kannadaonly 23. Let g: R→ R be a function such that, g() x=2 x + 5. 16. Thefunction f(), x= ex x∈ R is Then,whatisthevalueof g−1() x ? (a)ontobutnotone-one x − 5 (a) (b) 2x − 5 (b)one-oneonto 2 (c)one-onebutnotonto 5 x 5 (c) x − (d) + (d)neitherone-onenoronto 2 2 2 12 NDA/NA Mathematics

24. If A = {,,,}1 2 3 4 and R = {(,),(,),(,),(,),1 1 1 3 2 2 3 1 buy newspapers A, B and C, then the number of (,),(,),(,)}3 4 4 3 4 4 is a relation on AA× , then which families,whodonotbuyanynewspaperis oneofthefollowingiscorrect? (a)20 (b)80 (a) R isreflexive (c)0 (d)Noneofthese (b) R issymmetricandtransitive 34. Two finite sets A and B having m and n elements. (c) R istransitive,butnotreflexive The total number of relation from A to B is 64, then (d) R isneitherreflexivenortransitive possiblevaluesof m and n are 25. The function f: R→ R defined by f()() x= x2 + 1 35 for (a)2and4 (b)2and3 (c)2and1 (d)64and1 ∈ all x R is 35. Suppose AAA1,,..., 2 30 are thirty sets each having 5 (a)one-onebutnotonto elements and BBB1,,..., 2 n are n sets each having 3 30 n (b)ontobutnotone-one ∪ = ∪ = elements. Let ABSi j and each elements (c)Neitherone-onenoronto i =1 j = 1 (d)Bothone-oneandonto of S belongs to exactly 10 of A ’s and exactly 9 of B$ ’.s i j 26. Let N be the set of integers. A relation R on N is Thevalueof n isequalto defined as R={( x , y )| xy >0 , x , y , ∈ N }. Then, which (a)15 (b)3 oneofthefollowingiscorrect? (c)45 (d)Noneofthese (a) R issymmetricbutnotreflexive 36. Universal set, (b) R isreflexivebutnotsymmetric U={} x x5 −6 x 4 + 11 x 3 − 6 x 2 = 0 (c) R issymmetricandreflexivebutnottransitive (d) R isanequivalencerelation A={} x x2 −5 x + 6 = 0 27. Let R be the relation defined on the set of natural B={} x x2 −3 x + 2 = 0 number N as aRb; a, b∈ N , if a divides b. Then, Whatisthevalueof ()AB∩ ′ ? whichoneofthefollowingiscorrect? (a){1,3} (b){1,2,3} (c){0,1,3}(d){0,1,2,3} (a) R isreflexiveonly (b) R issymmetriconly 37. A relation R is defined on the set Z of integers as (c) R istransitiveonly follows mRn⇔ m + n isodd. (d) R isreflexiveandtransitive Whichofthefollowingstatementsis/aretruefor R? 28. Considerthefollowingstatements I. R isreflexive. II. R issymmetric. I. φ∈{} φ II. {}φ φ⊆ III. R istransitive. Select the correct answer using the code given below Which of the statements given above is/are correct? (a) IIonly (b)IIandIII(c)IandII (d)IandIII (a)Ionly (b)IIonly (c)BothIandII (d)NeitherInorII 38. Which of the following statements is not correct for the relation R defined by aRb if and only if b lives 29. Whichoneofthefollowingiscorrect? within1kmfrom a? (a) APAPA∪()() = (b) APAA∩() = − = − = (a) R isreflexive (c) APAA() (d) PAAPA(){}() (b) R issymmetric Here, P (A)denotesthepowersetofaset A. (c) R isnotanti-symmetric 30. If A, B and C are three finite sets, then [()]ABC∪ ∩ ′ (d)Noneoftheabove equalsto 39. Whatistheregionthatrepresents AB∩ ,if (a) ABC′ ∪ ′ ∩ ′ (b) ABC′ ∩ ′ ∩ ′ A={( x , y ) x + y ≤ }4 and B={( x , y ) x + y ≤ }0 ? (c) ABC′ ∩ ′ ∪ ′ (d) ABC∩ ∩ (a) {(,)}x y x+ y ≤ 2 (b) {(,)}x y2 x+ y ≤ 4 31. If A and B are two non-empty sets having n elements (c) {(,)}x y x+ y ≤ 0 (d) {(,)}x y x+ y ≤ 4 in common, then what is the number of common × × 40. Let X and Y be two non-empty sets and let R1 and R2 elementsinthesets AB and BA? be two relations from X into Y . Then, which one of 2 (a) n (b) n (c) 2n (d)Zero thefollowingiscorrect? =2 = =4 = ∆ ∩−1 ⊂−1 ∩ −1 32. If A{:} x x 1 and B{:} x x 1 , then AB is (a) ()RRRR1 2 1 2 ∩−1 ⊃−1 ∩ −1 equalto (b) ()RRRR1 2 1 2 − − − − − (a) {,}i i (b) {,}1 1 (c) ()RRRR∩1 =1 ∩ 1 (c) {,,,}−1 1 i − i (d)Noneofthese 1 2 1 2 ∩−1 =−1 ∪ −1 (d) ()RRRR1 2 1 2 33. There are 100 families in a society, 40 families buy newspaper A, 30 families buy newspaper B, 30 41. If a set A contains 4 elements, then what is the families buy newspaper C, 10 families buy numberofelementsin APA× ()? newspapers A and B, 8 families buy newspapers B (a)16 (b)32 and C, 5 families buy newspaper A and C, 3 families (c)64 (d)128 Sets,RelationsandFunctions 13

42. Let f: R→ R be a function defined as f( x )= x | x |; for 50. Therangeof R is each x∈ R, R being the set of real numbers. Which (a){3,6,10} (b){1,2} (c){2,3} (d){1,3} oneofthefollowingiscorrect? 51. Theinverserelation R −1 is (a) f isone-onebutnotonto (a){(6,2),(10,2),(3,3),(6,3),(10,5)} (b) f isontobutnotone-one (b){2,4} (c) f isbothone-oneandonto (c){(3,2),(1,3),(4,5)} (d) f isneitherone-onenoronto (d)Noneoftheabove 43. If A and B are subsets of a set X, then what is the valueof {()}AXBB∩ − ∪ ? Directions (Q. Nos. 52-55) Consider the ∪ ∩ universal set S = {0, 1,2, 3, 4, 5, 6, 7, 8, 9} and (a) AB (b) AB (c) A (d) B A = {,,,}1 2 3 4 , B = {2,3,5,6}, C = {2,3,7}, then = ∈ ≤ ≤ 44. Let U{:} x N1 x 10 be the universal set, N 52. Thevalueof A′ is = being the set of natural numbers. If A {,,,}1 2 3 4 (a){0,5,6,7,8,9} (b){2,3} = and B {,,,}2 3 6 10 , then what is the complement of (c){5,6} (d){7,8} ()AB− ? 53. Thevalueof ()AB− ′ is (a) {,}6 10 (b) {,}1 4 (a){0,2,3,5,6,7,8,9} (b){1,2,5} (c){2,3,5,6,7,8,9,10 (b){5,6,7,8,9,10} (c){2,6,7,8,9} (d)Noneofthese Directions (Q. Nos. 45-47) Each of these 54. Thevalueof AB′∩ is questions contain two statements, one is Assertion (A) (a){2,3} (b){5,6} (c){1,3} (d){1,4} and other is Reason (R). Each of these questions also has 55. Thevalueof BA′ − ′ is four alternative choices, only one of which is the correct (a){1,4} (b){2,3} (c){6,5} (d){4,2} answer. You have to select one of the codes (a), (b), (c) and (d) givenbelow. Directions (Q. Nos. 56-60) Read the following Codes passageandgiveanswer. (a)BothAandRareindividuallytrueandRisthe The students of a class are offered three languages correctexplanationofA. (Hindi, English and French). 15 students learn all the (b)BothAandRareindividuallytruebutRisnot three languages whereas 28 students do not learn any thecorrectexplanationofA. (c)AistruebutRisfalse. language. The number of students learning Hindi and (d)AisfalsebutRistrue. English but not French is twice the number of students learning Hindi and French but not English. The ∈2 < 45. Assertion (A) {x R | x 0 } is not a set. Here, R is number of students learning English and French but thesetofrealnumbers. not Hindi is thrice the number of students learning Reason(R) Foreveryrealnumber x, x2 ≥ 0. Hindi and French but not English. 23 students learn only Hindi and 17 students learn only English. The total number of students learning French is 46 and the 46. Assertion (A) If A = {1, 2, 3, 4}, then the relation R totalnumberofstudentslearningonlyFrenchis11. defined on A is {(1, 1), (2, 2), (3, 3), (4, 4)} is transitive. Reason (R) A relation R defined on A is transitive iff 56. Howmanystudentslearnpreciselytwolanguages? aRb, bRc ⇔ cRa ∀ a, b, c ∈A (a)55 (b)40 (c)30 (d)13 47. Assertion (A) If A = {1, 2, 3}, B = {2, 4}, then the 57. Howmanystudentslearn atleast twolanguages? numberofrelationfrom A to B isequalto26. (a)15 (b)30 (c)45 (d)55 Reason (R) The total number of relation from set A 58. Whatisthetotalstrengthoftheclass? ⋅ toset B isequalto {}2n()() A n B . (a)124 (b)100 (c)96 (d)66 59. HowmanystudentslearnEnglishandFrench? Consider a relation R Directions (Q. Nos. 48-51) (a)30 (b)43 (c)45 (d)73 is defined from a set A = {2, 3, 4, 5} to a set B = {3, 6, 7, 10} as follows (x, y) ∈R ⇔ x divides y. 60. Howmanystudentslearn atleast onelanguage? (a)45 (b)51 48. Express R, asasetoforderedpairsis (c)96 (d)Noneofthese (a){(2,4),(2,3)} 61. Foraset A,considerthefollowingstatements (b){(3,2),(3,7),(3,9)} I. APAPA∪()() = (c){(2,6),(2,10),(3,3),(3,6),(5,10)} ∩ = (d)Noneoftheabove II. {}()APAA III. PAAPA(){}()− = 49. Thedomainof R is Where P denotespowerset. (a){2,3,5} (b){1,2} Whichofthestatementsgivenaboveis/arecorrect? (c){2,3} (d)Noneofthese (a)Ionly (b)IIonly (c)IIIonly (d)I,IIandIII Answers LevelI 1. (a) 2. (d) 3. (b) 4. (c) 5. (d) 6. (d) 7. (d) 8. (c) 9. (b) 10. (c) 11. (d) 12. (b) 13. (b) 14. (d) 15. (d) 16. (d) 17. (d) 18. (c) 19. (a) 20. (a) 21. (a) 22. (a) 23. (d) 24. (d) 25. (d) 26. (d) 27. (d) 28. (b) 29. (a) 30. (a) 31. (d) 32. (d) 33. (b) 34. (a) 35. (d) 36. (c) 37. (a) 38. (c) 39. (a) 40. (b) 41. (d) 42. (c) 43. (b) 44. (c) 45. (c) 46. (c) 47. (b) LevelII 1. (d) 2. (d) 3. (a) 4. (c) 5. (c) 6. (b) 7. (c) 8. (b) 9. (c) 10. (d) 11. (d) 12. (a) 13. (b) 14. (c) 15. (d) 16. (c) 17. (d) 18. (d) 19. (c) 20. (d) 21. (d) 22. (d) 23. (a) 24. (d) 25. (c) 26. (d) 27. (d) 28. (d) 29. (c) 30. (c) 31. (b) 32. (a) 33. (a) 34. (b) 35. (c) 36. (c) 37. (a) 38. (c) 39. (c) 40. (d) 41. (c) 42. (c) 43. (a) 44. (c) 45. (a) 46. (a) 47. (a) 48. (c) 49. (a) 50. (a) 51. (a) 52. (a) 53. (a) 54. (b) 55. (a) 56. (c) 57. (c) 58. (a) 59. (a) 60. (c) 61. (b) Hints&Solutions LevelI 1. If (,),a b∈ A × B then a∈ A, b ∈ B,so a = 1 2, 11. Given that, there are two sets A and B, which contains 3 A = {,}1 2 and 6 elements each. ∴ ∪ = + − ∩ 2. (){,,,}{}ABAB∪ =1 2 3 4and ∩ = 3 Wehave, n()()()() A B n A n B n A B =3609[ + − = Q n () A=3 ,() n B = 6 ] ∴ ()()(,),(,),(,),(,)ABAB∪ × ∩ = {}1 3 2 3 3 3 4 3 or n() A∪ B =3 + 6 − 1 = 8 = ∩ = ∪ ∩ ⊆ ∪ 3. If AB, then ABAB, so ABAB. or n() A∪ B =3 + 6 − 2 = 7 4. n()()() A− B = n A − n A ∩ B =8 − 2 = 6 or n() A∪ B =3 + 6 − 3 = 6 5. The power set is a set of all subsets. So, it contain 2n [Q Theseare4possibilitiesof n()] A∩ B elements. 12. Totalnumberofstudents = 100 = = 6. Given, n(),() A4 n B 3 NumberofstudentshavingScience = 70 We know that, the total number of relations from two NumberofstudentshavingMathematics = 60 n()() A⋅ n B 4 × 3 12 finitesets A to B isgivenby =2 = 2 = 2 . Number of students having both Science and 7. If φ isanullset,thenitsotherrepresentationis{}. Mathematics = 40 i.e., φ = {}. Now,numberofstudentshavingScienceonly 8. Given, A= {,,} a b c =70 − 40 = 30 NumberofstudentshavingMathematicsonly Numberofsubsetsof A = 2n =23 = 8 =60 − 40 = 20 = = {where, n numberofelementsin A, n 3} Thus, number of students having Science only, ∴ Propersubsetof A =2n − 1 = 8 − 1 = 7 Mathematicsonlyandbothsubjects =30 + 20 + 40 = 90 9. Q A = {,,}1 2, 5 6 and B = {,}1 2, 3 Number of students, who have not taken Science Mathematicsbothofthesubjects ∴ × = AB {(,),(,),(,),(),(),(),11 12 13 21,,, 22 23 =100 − 90 = 10 (,),(,),(,)5 1 5 2 5 3 , (,),(,),(,)}6 1 6 2 6 3 13. Wehave, A = {,,,}1 2 3 4 and BA× = {(,),(,),(,),(,),(),()11 12 15 16 21,, 22 and R = {}(,),(,),(,),(,),(,),(,)11 13 22 23 31 32 (),(),(,),(,),(,),(,)}25,, 26 31 32 35 36 (i) R isnotreflexive,since (,)3 3 ∈R. ⇒ ()(){(,),(,),(,),()}ABBA× ∩ × = 1 1 1 2 2 1 2, 2 (ii) R issymmetric,sinceforeach (,)a b∈ R wehave (,)b a∈ R e.g., (,)(,),2 3∈RR ⇒ 3 2 ∈ 10. We have, ABC= ∪ …(i) (iii) R is not transitive, since for any (,)a b∈ R and ∴ EEEEEA−(((()))) − − − − (,),b c∈ R we do not find (,)a c∈ R e.g., (,),1 3 ∈R =EEEEA −((())) − − − c (,)3 2 ∈R but (,)1 2 ∉R. =EEEA −(())[] − − Q EAA−c = (iv) R is not anti-symmetric, since for any (,)a b∈ R and (,),b a∈ R wedonothave a= b. =EEA −() − c =EAA − = c [fromEq.(i)] e.g., (,),1 3 ∈R (,)3 1 ∈R but 1≠ 3. ⇒ ABCc=() ∪ c Hence, R isonlysymmetric. ∴ ABCc= c ∩ c Sets,RelationsandFunctions 15

14. Q A, B and C are non-empty sets, such that AC∩ = φ. 29. A ={,,,,}, −2 − 1 0 1 2 ∴ ()()()ABCBACB× ∩ × = ∩ ×=φ×=φB Since, R={( x , y ): y = | x |} 15. Q A={:}4 n + 2 n ∈ N ∴ −2RR 2, − 1 1 , 0RR 0, 1 1 and 2R 2 satisfies relation = {,,,,,,,}6 10 14 18 22 26 30 K y=| x |on A. = ∈ and B{:}3 n n N 30. A = {,,,},2 3 4 5 if a∈ A,then (,)a a∈ Rfor every a.So, Ris = {,,,,,,,,}3 6 9 12 15 18 21 24 K reflexive. ∴ ∩ = K AB {,,,}6 18 30 31. Q ABBC∪ = ∩ = + − ∈ {6 (n 1 ) 12 | n N } Fromabovestronginferenceis ABC⊆ ⊆ . = − ∈ {12n 6 | n N } e.g., A= {}, a B= {,}, a b C= {,,} a b c 16. Q AB∩ = φ (given) A∪ B = {,}, a b B∩ C = {,} a b ∴ ABAAB− = −()() ∩ Q ABA− = x − ′ = φ 32. Givenfunction, f() x = Now, BA x2 + 1 and BA∩ = φ Forthefunction f() x isone-one or BABA− ′ = ∩ f()() x= f x and ()ABBAB− ∩ = ∩ 1 2 ⇒ x1 = x2 17. Since, the sets A and B are not known, then cardinality x2 + 1x2 + 1 ∆ 1 2 oftheset AB cannotbedetermined. ⇒ 2 + =2 + x1 x 2 x1 x 1 x2 x 2 18. If A and B arefinitesets,then ⇒ − − − = x1 x 2()() x 1 x 2 x 1 x 2 0 n()()() A− B = n A − n A ∩ B ⇒ − − = ()()x1 x 2 x 1 x 2 1 0 Q = = ⇒ = ≠ 19. A {,,}1 2 3 and R {(,),(,),(,)}1 1 2 2 3 3 x1 x 2 and x1 x 2 1 Thisrelationisreflexive,since 1R 1, 2R 2 and 3R 3. So,thefunctionisone-one. x 20. Wehave, X = {multiplesof2} Let y = ⇒ yx2 + y = x Y = {multiplesof5}and Z = {multiplesof10} x2 + 1 ∴ XYZ∩() ∩ ⇒ x2 y− x + y = 0 =X ∩ {multiplesof5} ∩ {multiplesof10} ± − 2 1 1 4y 2 =X ∩ {multiplesof10} ⇒ x = ; 1− 4y ≥ 0 2y = {multiplesof2} ∩ {multiplesof10} ()()1− 2y 1 + 2 y ≥ 0 = {multiplesof10} = Z ()()2y− 1 2 y + 1 ≤ 0 ∴ XYZZ∩() ∩ = 21. Since, number of elements in X be n , then the number 2 +– + ofrelationson X and 2n . – ∞ –1/2 1/2 + ∞ 22. |x |<1 ⇒ − 1 < x < 1 (In this interval no natural number  1 1 willsatisfyit).So,itisanullset. y ∈ − , ~ {0 }  2 2 23. Q n() A = 3 and n() B = 2 ∴ = [()()]n A× n B  1 1 Numberofrelationsfrom A to B 2 Here,therangeof f( x )= − , ~ {0 } =2()3× 2 = 2 6 = 64  2 2 = 24. Since, intelligency is not defined for students in a class. and codomain R ⇒ Range ≠ Codomain i.e., Notawelldefinedcollection. ∴ f() x isnotonto. 25. AP=({,}){,{},{},{,}}1 2 = φ 1 2 1 2 Hence, f() x isone-onebutnotonto. Fromabove,itisclearthat 33. Given, A ={,,} −1 2, 5 8 and B = {,,,,}0 1 3 6 7 {,}1 2 ∈ A ∴ R ={(,), −1 0 (2,3),(5,6)} 26. Here, A and B are any two sets and U be the universal Q = set. ( RA isonelessthanfrom B) Hence,totalnumberofelementsin R is3. U − = − ∩ A B 34. Now, n()()() A B n A n A B ⇒ 47= 115 −n() A ∩ B ⇒ n() A∩ B = 68 ∴ ∪ = + − ∩ ∩ ∪ n()()()() A B n A n B n A B AAB() =115 + 326 − 68 = 373 27. A = {,,,},3 5 9 1 B = {,,,,,,,,}1 2 3 4 5 6 7 8 9 35. Thenumberofelementsinpowersetof A is1. Q =0 = Clearly, option (d) is not correct. [()]PA 2 1 ∴ PPA{()} =21 = 2 28. y= ex and y= e− x only common solution at (,)0 1 , so ⇒ =2 = AB∩ issingletonset. PPPA{{()}} 2 4 ⇒ PPPPA{{{()}}} =24 = 16 16 NDA/NA Mathematics

= ∈ ⇒ ∪ = ∪ = 36. Given, Na { ax | x N } A B{,,}, a b c A C{,,} a b c ∴ = K ∩ = ∩ = N12 {,,,,}12 24 36 48 A B{}, a A C{}, a = K ∴ For BC= and N 8 {,,,}8 16 24 ∴ ∩ = K ⇒ ABAC∪ = ∪ NN8 12 {,,}24 48 = and ABAC∩ = ∩ N 24 44. 37. From the given figure clearly, A Areaofshadedportion =()()ABAB ∪c + ∩ B Wearegiven |EAB |=42 ,| | = 15 ,| | = 12 and |(AB∪ )| = 22 Now,|(ABEAB∪ )|c = | | − |( ∪ )| =42 − 22 = 20 Also,weknowthat C |(ABABAB∪ )||||||( = + − ∩ )| ⇒|(ABABAB∩ )||||||( = + − ∪ )| =15 + 12 − 22 = 5 IntheaboveVenndiagram,shadedareashows ∴ Areaofshadedportion =20 + 5 = 25 ABC−() − . 38. Giventhat, n(), U = 700 n(), A = 200 n() B = 300 A B and n() A∩ B = 100 Weknowthat, n()()()() A∪ B = n A + n B − n A ∩ B =200 + 300 − 100 = 400 Now, n()()() A′ ∩ B ′ = n U − n A ∪ B C =700 − 400 = 300 39. Given, A = {1,4,9,16,25,36,49,64,81} IntheaboveVenndiagram,horizontallinesmean ()AB− andverticallinesmean ().AC∩ and B = {2,4,6,...} Totalshadedportion =()()ABAC − ∪ ∩ Now, AB∩ = {4,16,36,64} ∴ ()()()ABACABC− ∪ ∩ = − − ∴ Thecardinalityof(AB∩ ) = = = Numberofelementsin(AB∩ ) =4 45. n(),() T65 n C 84, n( T∪ C ) =120 and n ( T ∩ C ) = x 40. X={(4n − 3 n − 1 )| n ∈ N } Q n()()()() T∪ C = n T + n C − n T ∩ C = − ∈ and Y{9 ( n 1 )| n N } ⇒ 120≥ 65 + 84 − x, x ≥149 − 120 ≥ 29 ⇒ X = {,,,}0 9 54 K and x≤ n() T ⇒ x ≤ 65 and Y = {,,,,,,}0 9 18 27 36 54 K ∴ 29≤x ≤ 65 ∴ ∪ = = XY {,,,,,,...}0 9 18 27 36 54 Y 46. Q A and B are two sets satisfying ABBA− = − , which 41. The total number of elements common in ()AB× and ispossibleonly,if AB= . × 2 ()BA is n . (by property) 47. Let n(),() A= m n B = n 42. Total number of proper subsets of a finite set with n The total possible subsets of A and B are 2mand 2 n , =n − elements 2 1 (by property) respectively. 43. ∴ Giventhat, ABAC∪ = ∪ Accordingtothegivencondition, and ABAC∩ = ∩ 2m− 2 n = 56 Then,let A= {,}, a b B= {,} a c ⇒ 2n()() 2 m− n − 1 = 23 2 3 − 1 and C= {,} a c ⇒ n=3, m − n = 3 ⇒ m=6, n = 3 LevelII 1. Weknowthatbythedefinitionofsubsets,if AB= ∴Venndiagramrepresentsbothin(d). Then, ABABABX∪ = ∩,, ∀ ∈ 3. R is relation defined on NN× by (,)(,)a b R c d if and [()()()()]Q n A∪ B = n A + n B − n A ∩ B onlyif ad= bc, then R is ∈ andby De-Morgan’s law ()ABAB∪c = c ∩ c (i) Reflexive Since,forany a, b N ⇒ ab= ba ⇒ (,)(,)a b R a b 2. For first statement i.e., all poets (P) are learned people (L) and from second statement all poets who learned (ii) Symmetric Let (,)(,)a b R c d ⇒ ad= bc ⇒ bc= ad people(L)arehappy(H). ⇒ cb= da (Q commutativityofnaturalnumbers) ⇒ (,)(,)c d R a b P L L P L H (iii) Transitive Let (,)(,)a b R c d and (,)(,)c d R e f H ⇒ ad= bcand cf = de ⇒ ade= bce ⇒ acf= bce Sets,RelationsandFunctions 17

⇒ af= be (cancellationlaw) Symmetric ⇒ (,)(,)a b R e f Q 1R 2 but 2R 1 Hence, from above observations, we conclude that R is ∴ R isnotasymmetricrelation. anequivalencerelationon NN× . Transitive 4. AB= {,,,}1 2 3 4 and ={2,3,5} Q 1RRR 2, 2 3⇒ 1 3 Now, (,)x y = ordered pair of AB× , where x∈ A and ∴ R isatransitiverelation. y∈ B. 13. Giventhat, xSy isdefinedrelation, Theorderedpairs (,)x y suchthat x< y are if x2+ y 2 = 1 and ySx isdefinedrelation (1,2),(1,3),(1,5),(2,3),(2,5),(3,5),(4,5). 2 2 if y+ x = 1 Q = − 5. f( x ) cos2 x sin 2 x ⇒ x2+ y 2 = 1 2 2 2 2 [Q f( x )= a cos x + b sin x −a + b ≤ f()] x ≤ a + b Hence, xS y⇒ ySx −1 + 1 ≤cos 2x − sin 2 x ≤ 1 + 1 ∴Therelation S issymmetricrelation. −2 ≤cos 2x − sin 2 x ≤ 2 14. F() n = Setofalldivisorofanaturalnumber n. So,Rangeof f() x is [,].− 2 2 F (){,,,,},20= 1 2 4 5 10 F (){,,,}16= 1 2 4 8  1 1  Now, FF()(){,,}20∩ 16 = 1 2 4 6. Union of the closed sets 2 +,,,,...10 −  n = 1 2 is  n n  ∴ F(){,,} y = 1 2 4 (,).2 10 = setofalldivisorofanaturalnumber y 7. Q =sinθ + cos θ ∴ y = 4 =θ + θ Whichisoftheform Q asin b cos ,then 15. In the given figure marked region represents the people −a2 + b 2 ≤ Q ≤ a 2 + b 2 whocanspeakHindi,EnglishandKannadaonly. =x ∀ ∈ ⇒ −1 + 1 ≤Q ≤ 1 + 1 16. It is clear from the graph that f(), x e x R is one-one but not onto. Since, range ≠ Codomain, so f() x is ∴ −2 ≤Q ≤ 2 into. 8. The shaded portion represents in the given figure is y (())(()).PQRPRQ∩ − ∪ ∩ − y = ex 9. Since, on the set of real numbers, R is a relation defined 1 2 3 by xRyif and only if3x+ 4 y = 5 for which1R and R . x′ x 2 3 4 1 1 i.e., 1R ⇒ 3⋅ 1 + 4 ⋅ = 5, 2 2 y′ 2 3 2 3 and R ⇒ 3 × + ×4 = 5 3 4 3 4 17. Since, on taking a straight line parallel to x-axis, the groupofgivenfunctionintersectitatmanypoints. Hence,boththestatementsIIandIIIarecorrect. ∴ f() x ismany-one. Q 10. M = Set of men and R is a relation is son of defined on Andasrangeof f() x = Codomain M. ∴ f() x isonto. Reflexiverelation aRa. Hence, f() x ismany-oneonto. Since, a cannotbeasonof a. Symmetricrelation 18. An injective functionmeansone-one. = − aRa⇒ bRa Inoption(d), f(). x x Foreveryvaluesof x, wegetadifferentvalueof f. whichisalsonotpossible. Hence,itis injective. Transitiverelation aRb, aRc⇒ cRa 19. Q (,),(,),(,)a a b b c c∈ R whichisnotpossible. ∴ R isareflexiverelation. 2 3 11. Q A={:} n n ∈ N and B={:} n n ∈ N But (,)a b∈ R and (,).b a∉ R = K A {,,,,,,,,}1 4 9 16 25 49 64 81 ∴ R isnotasymmetricrelation. = K B {,,,,,}1 8 27 64 125 Also, (,),(,)a b b c∈ R AB∩ = {,,...}1 64 ⇒ (,)a c∉ R ∴ AB∩ mustbeapropersubsetof {:}m6 m∈ N ∴ R isnotatransitiverelation. 12. (a) Q R = {(,),(),(,),(,),(),(,)}11 22,, 33 12 23 13 20. Letusconsider a and b ∈()()α ∩ β × ξ ∩ η Reflexive ⇒ a ∈()α ∩ β and b ∈()ξ ∩ η Q 1RRR 1, 2 2, 3 3 ⇒ ()a ∈α and ()a ∈β and ()b ∈ξand (b ∈ η ) ∴ R isareflexiverelation. ⇒ (a∈α and b ∈ η ) and (a∈β and b ∈ ξ ) 18 NDA/NA Mathematics

⇒ ab ∈()α × η and ab ∈()β × ξ ⇒ a divides c ⇒ aRc ⇒ aRb, bRc⇒ cRa ∴ ab ∈()()α × η ∩ β × ξ Thus, R istransitive. ⇒ ()()()()α∩ β × ξ ∩ η = α × η ∩ β × ξ 28. Boththestatementsareincorrect. 21. In a Euclidean plane, orthogonality of lines is not an 29. APAA−() = equivalencerelation. Whichiscorrectas A and PA() aredisjointsets.

22. We define AEBc by the open sentence : A is cardinally 30. Weknowthat, equivalent to B on the family of sets. Then, the relation [()]()()ABCABCABC∪ ∩ ′= ∪ ′∪′= ′∩′∪′ Ec on family of set is an equivalence relation, which is = ′ ∩ ′ ∪ ′ true. ABC (byDe-Morgan’slaw) 31. Letusconsideranexample. 23. Let y=2 x + 5 Let A={,,} a b c and B={,,,} a b c d y − 5 −1 ⇒ y−5 = 2 x ⇒ x = = g() y Here,3elementsarecommonin A and B. 2 Now, − x − 5 ∴ g1 () x = × = 2 AB{(,),(,),(,),(,),(,),(,) aa ab acad ba bb , (,),(,),(,),(,),(,),(,)}bc bd ca cb cc cd 24. Since,3 ∈ A BA× ={(,),(,),(,),(,),(,),(,) aa ab acba bb bc , But (,)3 3 ∉R So,itisnotreflexive. (,),(,),(,),(,),(,),(,)}ca cb cc da db dc herecommonelementin ()AB× and ()BA× is9 i.e., and (,)3 4 ∈R and (,)4 3 ∈R (3)2. ∉ But (,)3 3 R So, in general, if A and B are two non-empty sets having So,itisalsonottransitive. ‘n’ elements in common, then (n)2 is the common Hence, R isneitherreflexivenortransitive. elementsinthesets AB× and BA× . 25. Since, f()()−1 = f 1 = 235 32. A={,},{,,,} −1 1 B = − 1 1 − i i i.e., tworealnumber1and–1havethesameimage. A− B =φ,{,} B − A = − i i So,thefunctionisnotone-oneandlet ∴ ()(){,}A− B ∪ B − A = − i i y=() x2 + 1 35 ⇒ x=() y /135 − 1 33. ∴nA()()()()()()∪ B ∪ C = nA + nB + nC − nAB − nBC Thus,everyrealnumberhasnopreimage.So,the −n()() CA + n ABC functionisnotonto. = 80 (numberoffamiliesreading atleast Hence,thefunctionisneitherone-onenoronto. onenewspapers A, B and C) Q = > ∈ 26. R{( x , y )| xy0 , x , y N } ∴ Totalnumberoffamilies = 100 Reflexive So,20familiesdonotreadanynewspaper. Q x, y∈ N 34. 2mn = 64 ⇒mn = 6 ∴ x, x∈ N ⇒ x2 > 0 ∴ ∴ R isreflexive. Thepossiblevalueof m and n are2and3. Symmetric 35. If elements are not repeated, then number of elements ∪ ∪ ∪ × Q ∈ in AAA1 2 ... n is 30 5. But each element is used x, y N 30× 5 > ⇒ > 10 times so, S = = 15. Similarly, if elements in and xy 0 yx 0 10 ∴ R isalsosymmetric. BBB1,,..., 2 n are not repeated, then total number of Transitive elementsis 3n buteachelementisrepeated9timesso, Q ∈ ⇒ > > ⇒ > 3n x,, y z N xy0, yz 0 xz 0 S = =15 ⇒n = 45. ∴ R isalsotransitive. 9 Thus, R isanequivalencerelation. 36. U={:} x x5 −6 x 4 + 11 x 3 − 6 x 2 = 0 = {,,,}0 1 2 3 27. For reflexive A={:} x x2 −5 x + 6 = 0 = {,}2 3 aRa⇒ a divides a and B={:} x x2 −3 x + 2 = 0 = {,}2 1 ∴ R isreflexive. AB∩ = { }2 Forsymmetric ∴ ()()ABUAB∩ ′ = − ∩ ⇒ aRb a divides b ={,,,}{}{,,}0 1 2 3 − 2 = 0 1 3 bRa⇒ b divides a 37. Q R is a relation defined on the set 2 of integers as whichmaynotbepossible. follows ∴ R isnotsymmetric. mRn⇔ m + n isodd. Fortransitive (I) We know that the sum of two odd and even numbers aRb⇒ a divides b⇒ b = ka isanevennumber.Thus,itmaybereflexiveornot. bRc⇒ b divides c⇒ c = lb (II) If m and n arenumbers,suchthat Now, c= lka mRn⇔ m + n isodd. Sets,RelationsandFunctions 19

Thus, nRm⇔ n + m isodd. ∴ (,)2 6 ∈R, (,)2 10 ∈R, (,)3 3 ∈R, (,)3 6 ∈R and ∈ ∴ Thisrelationissymmetric (,)5 10 R (III) Thisrelationisnottransitive. Thus, R = {(,),(,),(,),(,),(,)}26 210 33 36 510 Therefore,only(II)statementiscorrect. 49. Clearly,domain (){,,}R = 2 3 5 38. R isnotanti-symmetric. 50. Range (){,,}R = 3 6 10 39. Q A={}( x , y ) x + y ≤ 4 and B={}( x , y ) x + y ≤ 0 51. Bydefinition, R−1 = {(,),(,),(,),(,),(,)}62 102 33 63 105 ∴ A∩ B ={}( x , y ) x + y ≤ 0 52. A′ ={:}{,,,,,} x x ∈ Sand x ∉ A = 0 5 6 7 8 9 ∩−1 =−1 ∪ −1 40. Thecorrectrelationis ()RRRR1 2 1 2 . 53. A− B ={:}{,} x x ∈ Aand x ∉ B = 1 4 41. Since,thenumberofelementsinset A is4. ()(){,}ABSABS− ′ = − − = − 1 4 ∴ Numberofelementsin PA().=24 = 16 ={,,,,,,,}0 2 3 5 6 7 8 9 So,thenumberofelementsin APA× () =4 × 16 = 64. 54. AB′ ∩ ={,,,,,}{,,,}{,}056789 ∩ 2356 = 56 42. Q f( x )= x | x | 55. BSBS′ = − = −{,,,}{,,,,,}2 3 5 6 = 0 1 4 7 8 9 = If f()() x1 f x 2 ASAS′ = − = −{,,,}{,,,,,}1 2 3 4 = 0 5 6 7 8 9 ⇒ = x1| x 1 | x 2 | x 2 | ∴ BA′ − ′ ={,,,,,}{,,,,,}014789 − 056789 = {,}1 4 ⇒ x= x 1 2 Solutions (Q.Nos.56-60) ∴ f() x isone-one. ThetotalnumberofstudentslearningFrench = 46 Also,rangeof f() x = codomain of f() x ∴ f() x isonto. 23 17 Hence, f() x isbothone-oneandonto. H 2x E 43. Q AX⊆ and BX⊆ 15 x 3x ∴ {(())}AXBB∩ − ∪ ()Q XBB− = ′ =()()()ABBABBB ∩ ′ ∪ = ∪ ∩ ′ ∪ 11 F =()ABXAB ∪ ∩ = ∪ 28 44. Giventhat,U={:} x ∈ N1 ≤ x ≤ 10 ⇒ 15+ 11 +x + 3 x = 46 = = A {,,,}1 2 3 4 and B {,,,}2 3 6 10 ⇒ 4x = 20 ⇒ x = 5 Now, AB− ={,}1 4 56. The number of students learning precisely two ∴ Complementof(AB− ) languages =()()ABUAB − ′ = − − =x +2 x + 3 x ={,,,,,,,}2 3 5 6 7 8 9 10 =6x = 6 × 5 = 30 45. Both A and R are true and R is the correct explanation 57. The number of students learning atleast two languages ofA. =x +2 x + 3 x + 15 Q x2 isnevernegativebutitcanbepositiveorzero. =6x + 15 = 30 + 15 = 45 46. Since, {(1, 1), (2, 2), (3, 3), (4, 4)} is identity relation and 58. Thetotalstrengthoftheclass identity relation is always equivalence relation. So, it is =28 + 23 + 17 + 11 +x + 2 x + 3 x + 15 transitive. =79 + 6x + 15 = 79 + 30 + 15 = 124 Bydefinitionoftransitiverelation. 59. ThenumberofstudentslearnEnglishandFrench ⇔ ∀ ∈ aRb, bRc cRa, a,, b c A =15 + 3x =15 + 15 = 30 ∴ A and R are both correct and R is the correct 60. The number of students learn atleast one of the explanationofA. languages 47. We know by the property of relation, the total number of =124 − 28 = 96 relationfromset A toset B is 2n()() A⋅ n B . 61. Let A = {,}1 2 and {}{{,}}A = 1 2 So, Both A and R are true and R is the correct ⇒ = φ explanationofA. PA(){{},{},{,},},1 2 1 2 Then, APAPA∪()() ≠ 48. Recall that a/ b stands for ‘a divides b’. For the elements and {}(){{,}}{{},{},{,},}APA∩ =1 2 ∩ 1 2 1 2 φ of the given sets A and B, we find that 2/6, 2/10, 3/3, 3/6 = = and5/10. {,}1 2 A