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Chapter —5, , Countability and Metric space, , , , , Introduction, branch of mathematics, th, , >) we define a function and some of, es of countable and uncountabl, , in formulating the definitions of the me, , ce notion of a set and the concept of a fin,, its types We shall deal w, , , , rd exampl le sets in section (5.3), dstanee plays a vital role, , has limit of a function,, , , , convergence ofa sequence, contin, ance is called a “metric™. In sect, , < Sue, , pon. The generalization of dist, , tion of a metric., on of distance (metric) for real numbers is extended to abstrc, , yathematical structures are known as metric spaces Metric, , clul types of topological space. Metric spaces serve as be, study of position and) shapes including their propertic, , employ the concept of topology to model and understand, , id therefore topology has become an importan, , ay a major role in the mathematical field of analys, , |, pplications. Metric spaces were first introduced by Mau, , c in: mathematics in 1906. Section (3.4) of this chapter, , samples of metric spaces. In later sections, we sha, titi bo, , ivhbourhoods, open and closed sets, closure ane, , % Preliminaries

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Himalaya’s Mathematics B.Sc. (Sem V) (M10) Page 5.2, —S6-Se., age, , : ———_____, Relation:, , 1 and B be any two sets. A relation, , from a set A, product AX Bvi.e.,j i t, , ” ‘0 a set B is defined as a subset of the, / tion from A toBthen Rc 4 x B., , wewrite. xRy HT (Y) € R. xRy means ‘x ig R related to y° or ‘x is in the relation R to y., also. we write xRy iff (x,y) € Rand We say that x j, , ; % Is not R-related to y., Let R be a relation es asetA toaset B. The relation R- from B to A is Said to be the inverse, relation of R fF R™* = {(y,x): (x yy ep }., , Remarks: (1) Every relation has an inverse relation,, (2) A rel, , artesian, , ation on a set Aisa subset of the Cartesian product A x A., 3. Some types of relations:, , Arelation R in a set A is said to be:, , (i) Reflexive: iff xRx. vx EA (each element is related to itself), iff xRy => yRx, V(wy)E R,, (iii) Anti-symmetric: iff xRy , yRx => x =, , (iv) Transitive: iff xRy , yRz => xRz,, , (ii) Symmetric:, , y, VY) E R., , Vuy.zea., , Arelation R ina set A is called an equivalence relation iff it is reflexive, symmetric & transitive., 4. Equivalence class:, , Given an equivalence relation ‘~’, , on set A and x € A, we define a certain subset E of A by the, equation 7, , E={y€ A: y~x} or [x] ={yeA: y~x}, called as an equivalence class determined by x., Note: (1) x € [x]., (2) If x € fy] then [x] = [J]., (3) Two equivalence classes [x] & [y] are either disjoint or equal., , Partition:, , A Partition of a set A is a collection {A;} of disjoint subsets of A whose union is the full set 4, self, , The A,’s are called the partition sets., lquiy, , in PAI squivalence classes., ‘alence relation on the set A forms the partition of A into equivalence, Function:, , i ific. le : ynempty sets and R bea, function isa special case of relation. Vo be specific, let A & B be two nonemp, , 4 “ne B or it may relate an, ‘elation from A to B. then R may not relate an element of A to an element of, , slates each element of A to a unique, “Ment of 4 to more than one elements of B. Buta function relates each eleme i, ore than one, , “ment of B., , ae, , SN fT, , ae, , eek eee aml

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eS Soret &, Jas VEE, SULET eee at, , lonemMpty sets, , f tion trom A to B is aset f of ordered Pars in Ax, \ func trom ¢, € sts nique b € B with a,b) é f, , ha € A there exists a unique b with ( ; - |, ich that. lor each a € / . hoe |, 7 4 to B is defined to be a set f of ordered pairs nAx that, OR A function trom / s F B, , c B Xx B. ther =Cc. a, (a,b) & (a,c) belong to A x B. th b sof comes ., i | function f from a set A into a set B is a rule of cor spondence that, In other words. a fu id, , SSI As 4, , 3 i) B a 2, each clement x in A a uniquely determined element f(x) in, , i, , nent” is a function from A to B™ is usually represented Sy mbolically by f: A B., The statement ” is ¢ f, i respondence., A function is also called as mapping or map or Operation or corre P :, , A mapping of A into itself is sometimes called a transformation., , Wf: AB. then A is called the domain of f denoted by Dom f (oF D(f)). its membe, the first coordinates of the ordered pairs belonging to f and the set B is called the eoDomain A of f is also called the initial set of f and B the final set of f., , If (a,b) € f- itis customary to write h = f(a) ora — band we say that *b is the value off at, @ orb is the image of a under f * and a is the Pre-image of b*. The set Consisting of all the, , images of the elements of A under the function I is called the range of f or the image set of f, and is denoted by RCP) or F(A). Thus,, , RY) ={f@): xe, }., Note that, DPI=A & RU) CB., , Remark: From the, detined if, , TS are, domain,, , definition of a function itis clear that a map f AB is said to be well, (i) Any a@€A SS f(a) eB., , (ii) Any clement a € A>a unique element faye., , re than one E image,, than two elements of, , le. no clement of A can have mo}, (iii) lwo or more, , It is sometimes conveniet, , it to view a function as a, Produces a sin, , gle output if More than one Outputs, , ence between f& F(a)., ais the input give, , hich when given an input, rule is nota function., , j is analogous to, nto the Machine f, , are produced, the, There js, , 4 Vast differ, , : @ machine and f(a) is its, Praduct. In f(a) = b, y the Outputis b,, , 7. Some types of functions:, , Q) Let frAaB bea Map,, , (i) f iscalled 4 OnC-one (op injective, , )map it, re, or equivalently, %y # xs x, , 2 CaMV %2 GAs V(x, distinet Mages in B,, , ) = f(x): Ki», , MEAS Xp = XVD ¥ f(x), he., , distinct elements in A have

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Himalaya’s Mathematics B.Sc. (Sem y ) (M10) Page 5., . (NITO) Page 5.4, , | many-one map if distinct elements ind hav, , e, , th i ; ., unts of A have same image in B) © same f image in B (two or more, re 2 ;, , it $0) = £02) => x, ¥ X2 for some X1,xX2EA, , , called an onto (or surjective) map if f(A)=B, , ui) le. if every element in B has a pre, mage in A or every element of B is the image of some element in A, , ; js called an into map if there exist at least one ele, , + ment in B which is not the image of any, , element in A. Thus fis an into mapping if F(A) (range of f) is a proper subset of B, (codomain of f )., , wy A mapping which is one-one (injective) & onto (surjective) both, , is called a bijective map, (or one-to-one correspondence or equipollence )., , \n injective function is called an injection & surjective function is called surjection and a, junction which is one-one & onto both is called bijection., , (2) Inverse function:, , Consider a function f:A—>B. Let bEB be arbitrary. The inverse of the element b, denoted, oy f7'(b) is defined as a set Consisting of those elements in A which have 6 as their images., Thus, fT) ={aeA: fa)= b}., fB'isa subset of B then f-*(B’) = {aE A: faye BY CA,, 'tfisa one-one & onto map then f(b) will contain only one element of A. Thus,, , f-\(b) ={a@€A : f(a) =b}= a, where f(a) = b., , If f is one -one and onto then f-!(b) =a — f(a) =b. Thus, for every element, °€B. f~* corresponds to a unique element a € A such that f(a) = b showing thereby f~* is, amap,, , If f: A—B_ is one -one and onto map then f~? : B >A is also a map called an, verse map of f which is also one —one and onto., , '3) Equal functions:, , Thetieonings + An Band g: AB are said to be equal iff fx) = 90%), Vx € A and, Me f=g,, “Sof equal maps. the domains of mappings must be the same., , , , , ( r, , 4 entity function:, , Hee = ait r snes, Sach clement of A is mapped into itsell, then the map is called an identity map ind is denoted, ‘Tn (or iy), , 1 A, Int A A isan identity map given by Ly (x) =x. Vx E, , (6) ¢, Constant function:, yeER cor responds tloevery element, , Map fs . any element, ing a * ASB is called a constant function ifan clemen, , Pye R= y = f(x) VXEA., , "othe, ev Kar asingleton set., her Words, f is a constant function if the range of fis sing