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Telegram Channel : https://t.me/AIMSDARETOSUCCESS, Website : www.AIMSDARETOSUCCESS.blogspot.com, Subset &, Power Set, Equivalent set, Finite and set, Let A and B be two sets. If every element of A is an, element of B, then A is called a subset ofB and, written as AcB or BDA(read as 'A' is contained in, 'B' or 'B contains A). B is called superset of A., A set is a collection of well-defined distinct objects. The sets are, usually denoted by capital letters A, B, C etc., and the members or, elements of the set are denoted by lower case letters a, b, c etc.., If x is, a member of the set A, we write x e A (read as 'x belongs to A) and if, x is not a member of set A, we write x A(read as 'x doesn't belongs, to A). If x and y both belong to A, we write x, y e A., Some examples of sets are: A: odd numbers less than 10, Note:, The number of elements in a finite set is, 1. Every set is a subset and superset of itself., 2. If A is not a subset of B, we write AçB., 3. The empty set is the subset of every set., 4. Power Set: If A is a set with n (A) = m, then no. of, represented by n (A), known as cardinal, number., Eg.: A = {a, b, c, d, e} Then, n (A) = 5, N: the set of all natural numbers, V: the vowels in the English alphabates, R: the set of all real numbers., subset in power set n [P(A)]=2", e.g. Let A = {3, 4}, then subsets of A are o, {3}, {4},, {3,4}. Here, n(A) = 2 and number of subsets, of A = 2=4., In this form, we write a variable (say x) representing any member, of the set followed by property satisfied by each member of the set., e.g.: The set A of all prime number less than 10 in set builder form, is written as, A = {x | x is a prime number less than10}, The symbol "" stands for the word "such that". Sometimes, we use, symbol ":" in place of symbol "I", Introduction, Cardinal Number, Set builder form or Rule Method, Sets and, Representations-I, Kepresenta tion of Sets, Roaster or Tabular form, RE, In this form, we first list all the members of the set within braces, (curly brackets) and separate these by commas., e.g: The set of all odd natural number less than 10 in this form is, written as: A = {1,3,5,7,9}, In roaster form, every element of the set is listed only once., The order in which the elements are listed is immaterial., e.g. Each of the following sets denotes the same set, {1, 2, 3}, {3, 2,1}, {1, 3, 2}, Types of Sets, A set which has no element is called null set. It is denoted by, symbol o or {}., e.g: Set of all real numbers whose square is -1., In set-builder form: {x: x is a real number whose square is -1}, In roaster form: { } or o, Empty set or Null, set, A set which has finite number of elements is called a finite set., Otherwise, it is called an infinite set., e.g.: The set of all days in a week is a finite set whereas the set, of all integers, denoted by, {...-2, -1,0, 1, 2,...} or {x |x is integers} is an infinite set., An empty set o which has no element is a finite set is called, empty or void or null set., Singleton set, A set having one element is called singleton set., e.g.: (i) {0} is a singleton set, whose only member is 0., (ii) A = {x:1<x <3, x is a natural number} is a singleton set, which has only one member which is2, Two sets A and B are set to be equal, written as A=B, if every, element of A is in B and every element of B is in A., e.g.: (i) A = {1, 2, 3,4} and B = {3,1, 4, 2}, then A =B, (ii) A = {x : x-5% 0} and B = {x:xis an integral positive root, of the equation x-2x-15-0}, Then A= B, Two finite sets A and B are said to be equivalent, if, n(A) = n(B). Note: equal set are equivalent but equivalent, sets need not to be equal., e.g.: The sets A = {4,5, 3, 2} and B =, equivalent, but are not equal., Equal set, {1, 6, 8, 9} are
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Telegram Channel : https://t.me/AIMSDARETOSUCCESS, Website : www.AIMSDARETOSUCCESS.blogspot.com, omplement set, Symmetric, Union, Intersection, Subsets of a set, of real numbers 'R', Disjoint sets, A Venn diagram is an illustration of the relationships between, The set containing all objects of element and of, which all other sets are subsets is known as, and among sets, groups of objects that share something common., These diagrams consist of rectangle and closed curves usually, In the given venn diagram, U={1,2,3,.10} universe, 3 set of which A={2,4,6,8,10}, and B={4,6} are subsets, universal sets and denoted by U., e.g : For the set of all intergers, the universal, set can be the set of rational numbers or the set, R of real numbers, circles, e.g시, Venn Diagram, 6., Universal Set, 1. For any set A, we have, (a) AUA=A, (b) AnA=A, (c) AU=A, (d) And=0, (e) AU U=U, (f) AnU=A, (g) A- 0=A, (h)A-A=0, 2. For any two sets A and B we have, (a) AUB = BUA, (b) AnB= BnA, (c) A-BCA, (d) B-ACB, 3. For any three sets A,B and C, we have, (a) AU(BUC)= (AUB)UC, (b) An(BnC)=D (AnB)nC, (c) AU(BnC)=(AUB)N(AUC), (d) An(BUC)=(AnB) U(AnC), (e) A-(BUC)=(A-B) )(A-C), (f) A-(BnC)=(A-B)U(A-C), Algebra of sets, Sets, Relations and, Functions Part-II, Operations on Sets, Difference of two sets, The symmetric difference of two sets, A and B, denoted by A A B, in defined, as (A A B)=(A-B) U (B-A), e.g. If A = {1,2,3,4,5}, and B = {1,3,5,7,9} then, (A A B)= (A-B) U (B-A), = {2, 4,} U {7,9}, = {2, 4, 7,9}, If A and B are two sets, then their, difference A-B is defined as:, A-B = {x: x e A and x B}, Similarly, B- A = {x:x eB and x £A}, e.g. If A={1,2,3,4,5,} and B={1,3,5,7,9}, then, A-B={2,4} and B-A={7,9}, The union of two sets A and B, written as AUB (read as A union, B) is the set of all elements which are either in A or in B or in, both. Thus, AUB = {x:x EA or x eB}, clearly, x e AUB= x eA or x e B and x ¢AUB=x ¢A and x eB, eg: If A = {a, b, c, d} and B = {c, d, e,f}, then AUB = {a, b, c, d, e, f}, AUB, The intersection of two sets A and B, written as AnB (read as, 'A' intersection 'B') is the set consisting of all the common, elements of A and B., Thus, AnB = {x: xeAand xeB}, Clearly, xeAnB= {xeA and xeB} and xeAnB = {x ¢A or, •The set of natural numbers, N={1,2, 3, 4, 5,---}, • The set of integers, Z={...-3,-2,1,0, 1, 2, 3,---}, • The set of irrational numbers,, T = {x: xeR and xeQ}, • The set of rational number, If U is a universal set and A is a subset, of U, then complement of A is the set, which contains those elements of U,, which are not present in A and is, denoted by A' or A". Thus,, AC = {x: xeU and xgA}, e.g.: If U = {1,2, 3, 4, ...}, and A = {2, 4, 6, 8, ....}, 7= x: x}=0, P, q eZ and q+ 0}, Relation among these subsets arethen AC = {1, 3,5, 7,...}, NCZCQ,QcR,TcR, NaT, AnB, A, Properties of complement, e.g: If A={a,b,c,d} and B={c,d,e,f}, Then AnB = {c,d}, Interval Notation, Let a and b be real numbers with a <b, Complement law:, Region on the, real number line, Set of Real Numbers, Interval Notation, • De morgan's Law:, (i) (AUB)' = A' B' (ii) (AB)' = A'UB', Double Complement law:, (A')' =A, • Law of empty set and universal set, d'=U and U'=, (a, b), [a, b), (a, b], {4 > x >0|x}•, Two sets A and B are said to be disjoint, if AB =6i.e, A and B, have no common element. e.g: if A = {1, 3,5} and B = {2,4, 6}, Then, AnB =, so A and B are disjoint., {q > xs미자., [a, b], (-00, b), {q>x>비x}•, {9 > x|x, {95x, {qsxs미x}, {v< x\x}, {vz x|x} •, [q 'o0-), (00 'e), (00 'e], (0 'o0-), R
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Telegram Channel : https://t.me/AIMSDARETOSUCCESS, Transitive relation, Composition of, Invertible funct, Types of Relations, lypes of, Functions., le functions, Website : www.AIMSDARETOSUCCESS.blogspot.com, The composition of functions f: A B, and g: B C is denoted by gof, and is, defined as gof : A C given by gof(x) :, g(f(x)) VXEA e.g. let A = N and f, g :, N →N such that f(x) =x and g(x) = x, V xeN. Then gof(2), = 4° = 64., A function f : X Y is invertible, if 3 a function, g: Y X such that gof = Ix and fog = ly. Then,, g is the inverse of f. If f is invertible, then it is both, Reflexive relation, Symmetric relation, A relation R: A→A is, Invertible, reflexive if aRa V aeA, one-one and onte, and vice-versa., eg. If f(x) =x and f: N N, then f is invertible., Theorem 1: If f:X Y,g: Y Z and H: Z→S, A relation R: A → A is symmetric, are functions, then ho(gof), if aRb + bRa va, be A, (hog)of., Theorem 2: Let f : X Y and g : Y Z be two, invertible functions, then gof is invertible with, A relation R : A A is transitive, if aRb and bRc = aRc V a,b,cEA., Sets, Relations and, Functions Part-III, Equivalence relation, (If a relation has reflexive, symmetric, and transitive relations) e.g., Let T =, the set of all triangles in a plane and, R:T T defined by R = {(T, T)} : T,, is congruent to T,}.Then, R is, One-one (injective), (surjective), f:X→Y is onto if e, Onto, f:X→Y is one-one if, , x, e X. Other wise,, fis many- one,, every, ye Y,3 xeX such that f(x) = y,, Then fis surjective, f:X-Y is both one-one, and onto, then fis bijective., ex = 'x = ("x)f = (x), f, is one-one., f, is onto. f, f, is bijective., SSHOnS OI
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Telegram Channel : https://t.me/AIMSDARETOSUCCESS, Inverse relation, Even and, dd function, Kinds of Functions, Some standard real functions, Website : www.AIMSDARETOSUCCESS.blogspot.com, Cartesian product of sets, Pictorial representation of, Given two non empty sets A&B. The cartesian, product AxB is the set of all ordered pairs of, elements from A&B i.e., AxB={(a,b): a e A; be B}., If n(A)3p and n(B)%3Dq, then n(AxB)3Dpq, One-One Onto function, One-One Into function, Definition: A relation 'f' from a set A to set B is said to be a, function if every element of set A has one, and only one image in set B., Notations:, Domain(Input) L, A-, - y=f(x) Range(Output), Let A&B be two non- empty sets. Then any subset, 'R'of Ax B is a relation from A to B.If (a,b) eR, then, we write a R b, which is read as 'a is related to b', by a relation R, 'b' is also called image ofa' under R., The total number of relations that can be defined, from a set A to a set B is the number of possible, subsets of A x B. If n(A)=p and n(B)3Dq, then, n (A x B)3Dpq and total number of relations is 2"., Range of f, Range=Codomain, Domain of 'f', Many-One Onto function, Codomain of 'f, Given, R={(x,y): x is the first letter, of the name y, x eP, y eQ}, Then, R={(a, Ali), (b, Beena),, Bindu (C, charu)}This is a visual or, fOChariss pictorial representation of relation, Dirya/ R (called an arrow diagram)is, shown in figure., В, Range=Codomain, RangecCodomain, IVe, Beena, Let f:xR and g:xR be any two real functions, where xcR., Addition: (f+g) x = f(x)+g(x); VxeR, Subtraction: (f-g) x f(x)-g(x); V xe R, Product: (fg) x = f(x).g(x); V x e R, Quotient: (f/g) (x) = f(x)/g(x); provided, g(x) #0, Vx e R, Relation, Sets, Relations, and, Functions Part-IV, If R is a relation from A to B, then the set of first, elements in R is called domain & the set of second, elements in R is called range of R. Symbolically,, Domain of R={x:(x,y) e R}; Range of R={y:(x,y) e R}, The set B is called co-domain of relation R., Note: the range c Codomain., e.g. Given, R={(1,2),(2,3),(3,4),(4,5),(5,6)},, Algebra, lof functions, a relation, f(x)=log,x, a>0, a +1 Domain=x e (0, 0) Range%3Dy e R Log function, The function f.R →R defined by y=f(x)%3Dx V x eR is called, identity function. Domain=R and Range=R, The function f.R R defined by y=f(x)3Dc, V x e R, where, c is a constant, is called constant function. Domain=R\, and Range {c}, Domain &, Identity, function, of a Relation then Domain of R={1,2,3,4,5}, range, Constant, function, codomain of R={1,2,3,4,5,6}, The functionf:R →R defined by f(x)= [x; x 20, is called modulus function., It is denoted by y=f(x)%3D[x|. Domain=R and Range (0, co), The function f:RR defined by f(x)= [1x>0] is called signum Signum, function. It is usually denoted by, y=f(x)%3Dsgn(x) Domain=R and Range ={-1,0, 1}, The function f:R R defined by as the greatest integer, less than or equal to x .It is usually denoted by y=f(x)%3D[x].integer function Exponential f(x)=ax, a>0, a +1,, Domain=R and Range=Z(All integers), Let A&B be two sets and R be a relation from set A, to set B.Then inverse of R,denoted by R-1, is a, relation from B to A and is defined by, R-1={(b,a) : (a,b) e R}. Note: (a,b) eR (b,a) e R-1, Also, Dom(R)=Range (R-1) and Range(R)=Dom(R-1), Modulus, function, Even function, So >x x-1, f(-x)%3Df(x), Vxe Domain, Odd function, 0 = x'0, (o >x'I-, function, f(-x)=-f(x), Vxe Domain, Greatest, function, Domain: xeR: Range:f(x)e(0,00)
