Set, Relations and Functions Notes

Page # 1 SET, RELATIONS AND FUNCTIONS SET, RELATIONS AND FUNCTIONS 1. SET THEORY Collection of well defined objects which are distinct and distinguishable. A collection is said to be well defined if each and every element of the collection has some definition. Notation of a set : Sets are denoted by capital letters like A, B, C or { } and the entries within the bracket are known as elements of set. Cardinal number of a set : Cardinal number of a set X is the number of elements of a set X and it is denoted by n(X) e.g. X = [x1,x2,x3] n(X) =3 2. REPRESENTATION OF SETS Set Listing Method (Roster Method) : In this method a set is described by listing all the elements, separated by commas, within braces { } Set builder Method (Set Rule Method) : In this method, a set is described by characterizing property P(x) of its elements x. In such case the set is described by {x : P(x) holds} or {x | P(x) holds}, which is read as the set of all x such that P(x) holds. The symbol ‘|’ or ‘:’ is read as such that. 3. TYPE OF SETS Finite set : A set X is called a finite set if its element can be listed by counting or labeling with the help of natural numbers and the process terminates at a certain natural number n. i.e. n(X) = finite no. eg (a) A set of English Alphabets (b) Set of soldiers in Indian Army Infinite set : A set whose elements cannot be listed counted by the natural numbers (1, 2, 3.......n) for any number n, is called a infinite set. e.g. (a) A set of all points in a plane (b) X = {x : x  R, 0 < x < 0.0001} (c) X = {x : x  Q, 0  x  0.0001} Singleton set : A set consisting of a single element is called a singleton set. i.e. n (X) = 1, e.g. {x : x  N, 1 < x < 3}, {{}} : Set of null set, {} is a set containing alphabet . Null set : A set is said to be empty, void or null set if it has no element in it, and it is denoted by . i.e. X is a null set if n(X) = 0. e.g. : {x : x  R and x2 + 2 = 0}, {x : x > 1 but x < 1/2}, {x : x  R, x2 < 0} Equivalent Set : Two finite sets A and B are equivalent if their cardinal numbers are same i.e. n(A) = n(B). Equal Set : Two sets A and B are said to be equal if every element of A is a member of B and every element of B is a member of A. i.e. A = B, if A and B are equal and A  B, if they are not equal. 4. UNIVERSAL SET It is a set which includes all the sets under considerations i.e. it is a super set of each of the given set. Thus, a set that contains all sets in a given context is called the universal set. It is denoted by U. e.g. If A = {1, 2, 3}, B = {2, 4, 5, 6} and C = {1, 3, 5, 7}, then U = {1, 2, 3, 4, 5, 6, 7} can be taken as the universal set. 5. DISJOINT SET Sets A and B are said to be disjoint iff A and B have no common element or A  B = . If A  B   then A and B are said to be intersecting or overlapping sets. e.g. : (i) If A = {1, 2, 3}, B = {4, 5, 6} and C = {4, 7, 9} then A and B are disjoint set where B and C are intersecting sets. (ii) Set of even natural numbers and odd natural numbers are disjoint sets. 6. COMPLEMENTARY SET Complementary set of a set A is a set containing all those elements of universal set which are not in A. It is denoted by A , AC or A. So AC = {x : x  U but x  A}. e.g. If set A = {1, 2, 3, 4, 5} and universal set U = {1, 2, 3, 4,.......50} then A = {6, 7, .......50} NOTE : All disjoint sets are not complementary sets but all complementary sets are disjoint. 7. SUBSET A set A is said to be a subset of B if all the elements of A are present in B and is denoted by A  B (read as A is subset of B) and symbolically written as : xAxBAB Page # 2 SET, RELATIONS AND FUNCTIONS U Number of subsets : B Consider a set X containing n elements as {x1, x2, ......., xn} then the total number of subsets A of X = 2n e.g. If A is a set then the complement of A is Proof : Number of subsets of above set is equal represented in fig. U to the number of selections of elements taking A any number of them at a time out of the total n A elements and it is equal to 2n  C0 + nC1 + nC2 + ...... + nCn = 2n n 10. OPERATIONS ON SETS Types of Subsets : A set A is said to be a proper subset of a set B if Union of sets : every element of A is an element of B and B has at If A and B are two sets then union () of A and B least one element which is not an element of A is the set of all those elements which belong either and is denoted by A  B. to A or to B or to both A and B. It is also defined The set A itself and the empty set is known as as A  B = {x : x  A or x  B}. It is represented improper subset and is denoted as through Venn diagram in fig.1 & fig.2 A  B. U e.g. If X = {x1, x2, .....,xn} then total number of proper sets = 2n – 2 (excluding itself and the null A U B A B set). The statement A  B can be written as B  A, then B is called the super set of A and is written as B  A. 8. POWER SET The collection of all subsets of set A is called the power set of A and is denoted by P(A) i.e. P(A) = {x : x is a subset of A}. I f X = {x1, x2, x3, ..........xn} then n(P(X)) = 2n ; n(P(P(x))) = 2 2n . Fig. (1) Fig. (2) Intersection of sets : If A and B are two sets then intersection () of A and B is the set of all those elements which belong to both A and B. It is also defined as A  B = {x : x  A and x  B} represented in Venn diagram (see fig.) U 9. VENN (EULER) DIAGRAMS The diagrams drawn to represent sets are called A B Venn diagram or Euler-Venn diagrams. Here we represents the universal U as set of all points within rectangle and the subset A of the set U is represented by the interior of a circle. If a set A is a subset of a set B, then the circle representing A is drawn inside the circle representing B. If A and B are not equal but they have some common elements, then to represent A and B by two intersecting circles. e.g. If A is subset of B then it is represented diagrammatically in fig. Difference of two sets : If A and B are two sets then the difference of A and B, is the set of all those elements of A which do not belong to B. U A B A–B Thus, A – B = {x : x  A and x  B} or A – B = {x  A ; x  B} Clearly x  A – B  x  A and x  B It is represented through the Venn diagrams. Page # 3 SET, RELATIONS AND FUNCTIONS Symmetric difference of two sets : Set of those elements which are obtained by taking the union of the difference of A & B is (A – B) & the difference of B & A is (B – A), is known as the symmetric difference of two sets A & B and it is denoted by (A  B). Thus A  B = (A – B)  (B – A) Representation through the venn diagram is given in the fig. U A B AB 11. NUMBER OF ELEMENTS IN DIFFERENT SETS If A, B & C are finite sets and U be the finite universal set, then (i) n(AB) = n(A) + n(B) – n(A B) (ii)n(AB) = n(A) + n(B) (if A & B are disjoint sets) (iii) n(A – B) = n(A) – n(A B) (iv) n(A B) = n[(A – B)  (B – A)] = n(A) + n(B) – 2n(A  B) (v) n(A  B  C) = n(A) + n(B) + n(C) – n(A B) – n(B  C) – n(A  C) + n(A  B  C) (vi) n(A B) = n(A  B) = n(U) – n(A  B) (vii) n(A B) = n(A  B) = n(U) – n(A  B) 12. CARTESIAN PRODUCT OF TWO SETS Cartesian product of A to B is a set containing the elements in the form of ordered pair (a, b) such that a  A and b  B. It is denoted by A × B. i.e. A × B = {(a, b) : a  A and b  B} = {(a1, b1),(a1, b2),(a2, b1),(a2, b2),(a3, b1),(a3, b2)} If set A = {a1, a2, a3} and B = {b1, b2} then A × B and B × A can be written as : A × B = {(a, b) : a  A and b  B and B × A = {(b, a) ; b  B and a  A} Clearly A × B  B × A until A and B are equal Note : 1. If number of elements in A : n(A) = m and n(B) = n then number of elements in (A × B) = m × n 2. Since A × B contains all such ordered pairs of the type (a, b) such that a  A & b  B, that means it includes all possibilities in which the elements of set A can be related with the elements of set B. Therefore, A × B is termed as largest possible relation defined from set A to set B, also known as universal relation from A to B. 13. ALGEBRAIC OPERATIONS ON SETS Idempotent operation : For any set A, we have (i) A  A = A and (ii) A  A = A Proof : (i) A  A = {x : x  A or x  A} = {x : x  A}=A (ii) A  A = {x : x  A & x  A} = {x : x  A}=A Identity operation : For any set A, we have (i) A   = A and (ii) A  U = A i.e.  and U are identity elements for union and intersection respectively Proof : (i) A   = {x : x  A or x  } = {x : x  A} = A (ii) A  U = {x : x  A and x  U} = {x : x  A} = A Commutative operation : For any set A and B, we have (i) A  B = B  A and (ii) A  B = B  A i.e. union and intersection are commutative. Associative operation : If A, B and C are any three sets then (i) (A  B)  C = A  (B  C) (ii) (A  B)  C = A  (B  C) i.e. union and intersection are associative. Distributive operation : If A, B and C are any three sets then (i) A  (B  C) = (A  B)  (A  C) (ii) A  (B  C) = (A  B)  (A  C) i.e. union and intersection are distributive over intersection and union respectively. De-Morgan’s Principle : If A and B are any two sets, then (i) (A  B) = A  B (ii) (A  B) = A  B Proof : (i) Let x be an arbitrary element of (A  B) . Then x  (A  B)  x  (A  B)  x  A and x  B  x  A  B Again let y be an arbitrary element of A  B. Then y  A  B  y  A and y  B  y  A and y  B  y  (A  B)  y  (A  B)  A  B  (A  B). Hence (A  B) = A  B Similarly (ii) can be proved. 14. RELATION A relation R from set X to Y (R : X  Y) is a correspondence between set X to set Y by which some or more elements of X are associated with some or more elements of Y. Therefore a relation Page # 4 SET, RELATIONS AND FUNCTIONS (or binary relation) R, from a non-empty set X to another non-empty set Y, is a subset of X × Y. i.e. RH : X  Y is nothing but subset of A × B. e.g. Consider a set X and Y as set of all males and females members of a royal family of the kingdom Ayodhya X = {Dashrath, Ram, Bharat, Laxman, shatrughan} and Y = {Koshaliya, Kakai, sumitra, Sita, Mandavi, Urmila, Shrutkirti} and a relation R is defined as “was husband of ”from set X to set Y. RH Dashrath Ram Bharat Laxman Shatrughan X Koshalya Sumitra Kakai Urmila Sita Mandavi Shrutkirti Y Then RH = {(Dashrath, Koshaliya), (Ram, sita), (Bharat, Mandavi), (Laxman, Urmila), (Shatrughan, Shrutkirti), (Dashrath, Kakai), (Dashrath, Sumitra)} Note : (i) If a is related to b then symbolically it is written as a R b where a is pre-image and b is image (ii) If a is not related to b then symbolically it is written as a R / b. Domain, Co-domain & Range of Relation Domain : of relation is collection of elements of the first set which are participating in the correspondence i.e. it is set of all pre-images under the relation R. e.g. Domain of RH : {Dashrath, Ram, Bharat, Laxman, Shatrughan} Co-Domain : All elements of set Y irrespective of whether they are related with any element of X or not constitute co-domain. e.g. Y= {Koshaliya, Kakai, Sumitra, Sita, Mandavi, Urmila, Shrutkirti} is co-domain of RH. Range : of relation is a set of those elements of set Y which are participating in correspondence i.e. set of all images. Range of RH : {Koshaliya, Kakai, Sumitra, Sita, Mandavi, Urmila, Shrutkirti}. 15.TYPES OF RELATIONS Reflexive Relation  R : X  Y is said to be reflexive iff x R x V x  X. i.e. every element in set X, must be a  x  X; (x, x)  R then related to itself therefore V relation R is called as reflexive relation. Identity Relation : Let X be a set. Then the relation Ix = {(x, x) : x  X} on X is called the identity relation on X. i.e. a relation Ix on X is identity relation if every element of X related to itself only. e.g. y = x Note : All identity relations are reflexive but all reflexive relations are not identity. Symmetric Relation R : X  Y is said to be symmetric iff (x, y)  R  (y, x)  R for all (x, y)  R i.e. x R y  y R x for all (x, y)  R. e.g. perpendicularity of lines in a plane is symmetric relation. Transitive Relation R : X  Y is transitive iff (x, y)  R and (y, z)  R  (x, z)  R for all (x, y) and (y, z)  R. i.e. x R y and y R z  x R z. e.g. The relation “being sister of ” among the members of a family is always transitive. Note : (i) Every null relation is a symmetric and transitive relation. (ii) Every singleton relation is a transitive relation. (iii) Universal and identity relations are reflexive, symmetric as well as transitive. Anti-symmetric Relation Let A be any set. A relation R on set A is said to be an antisymmetric relation iff (a, b)  R and (b, a)  R  a = b for all a, b  A e.g. Relations “being subset of ”; “is greater than or equal to” and “identity relation on any set A” are antisymmetric relations. Equivalence Relation A relation R from a set X to set Y (R : X  Y) is said to be an equivalence relation iff it is reflexive, symmetric as well as transitive. The equivalence relation is denoted by ~ e.g. Relation “is equal to” Equality, Similarity and congruence of triangles, parallelism of lines are equivalence relation. 16. INVERSE OF A RELATION Let A, B be two sets and let R be a relation from a set A to B. Then the inverse of R, denoted by R–1 , is a relation from B to A and is defined by R–1 = {(b, a) : (a, b)  R}, Clearly, (a, b)  R  (b, a)  R–1 Also, Dom of R = Range of R–1 and Range of R = Dom of R–1 Page # 5 SET, RELATIONS AND FUNCTIONS 1. FUNCTION If to every value (Considered as real unless other wise stated) of a variable x, which belongs to some collection (Set) E, there corresponds one and only one finite value of the quantity y, then y is said to be a function (Single valued) of x or a dependent variable defined on the set E ; x is the argument or independent variable . If to every value of x belonging to some set E there corresponds one or several values of the variable y, then y is called a multiple valued function of x defined on E.Conventionally the word "FUNCTION” is used only as the meaning of a single valued function, if not otherwise stated. x Pictorially :    input (ii) output (iii)  B satisfies the (ii) DOMAIN, CODOMAIN & RANGE OF A FUNCTION : Let f : A  B, then the set A is known as the domain of f & the set B is known as co-domain of f . The set of all f images of elements of A is known as the range of f . Thus : Domain of f = {a  a  A, (a, f(a))  f}  A, f(a)  B} It should be noted that range is a subset of co (IV) 3. (i) IMPORTANT TYPES OF FUNCTIONS : POLYNOMIAL FUNCTION : If a function f is defined by f (x) = a0 xn + a1 xn + a2 xn2 + ... + an1 x + an where n is a non negative integer and a0, a1, a2, ..., an are real numbers and a0  0, then f is called a polynomial function of degree n . NOTE : (a) A polynomial of degree one with no constant term is called an odd linear 1 where  0. EXPONENTIAL FUNCTION : A function f(x) = ax = ex ln a (a > 0 , a  1, x  R) is called an exponential function. The inverse of the exponential function is called the logarithmic function . i.e. g(x) = loga x . (v) Range of f = {f(a)  a domain . If only the rule of function is given then the domain of the function is the set of those real numbers, where function is defined. For a continuous function, the interval from minimum to maximum value of a function gives the range. g( x ) , h (x ) g (x) & h (x) are polynomials & h (x) a 2. FRACTIONAL RATIONAL FUNCTION : A rational function is a function of the form. y = f (x) =   A  (a, f(a))  f and (iii) (a, b)  f & (a, c)  f  b = c ALGEBRAIC FUNCTION : y is an algebraic function of x, if it is a function that satisfies an algebraic equation of the form P0 (x) yn + P1 (x) yn1 + ....... + Pn1 (x) y + Pn (x) = 0 Where n is a positive integer and P0 (x), P1 (x) ........... are Polynomials in x. e.g. y = x is an algebraic function, since it satisfies the equation y²  x² = 0. Note that all polynomial functions are Algebraic but not the converse. A function that is not algebraic is called TRANSCEDENTAL FUNCTION . f (x )  y   , y is called the image of x & x is the pre-image of y under f. Every function from A following conditions . (i) f AxB function . i.e. f(x) = ax , a  0 (b) There are two polynomial functions , satisfying the relation ; f(x).f(1/x) = f(x) + f(1/x). They are : (i) f(x) = xn + 1 & (ii) f(x) = 1  xn , where n is a positive integer . ABSOLUTE VALUE FUNCTION : A function y = f (x) = x is called the absolute value function or Modulus function. It is defined as : y = (vi)  x if x  0   x if x  0 x  SIGNUM FUNCTION : A function y= f (x) = Sgn (x) is defined as  1 for x  0  0 for x  0 follows : y = f (x) =    1 for x  0 (vii) It is also written as Sgn x = |x|/ x ; x  0 ; f (0) = 0 GREATEST INTEGER OR STEP UP FUNCTION : The function y = f (x) = [x] is called the greatest integer function where [x] denotes the greatest integer less than or equal to x . Note that for : 1  x < 0 ; [x] =  1 0 Page # 6 SET, RELATIONS AND FUNCTIONS x< 1 1x< 2 2x < 3 ; [x] = 0 Into function : ; [x] = 1 If f : A ; [x] = 2  B is such that there exists atleast one element in codomain which is not the and so on . (viii) FRACTIONAL PART FUNCTION : It is defined as : g (x) = {x} = x  [x] . e.g. the fractional part of the no. 2.1 is 2.1 2 = 0.1 and the fractional part of  3.7 is 0.3. The period of this function is 1 and graph of this function is as shown . 4. image of any element in domain, then f(x) is into . Thus a function can be one of these four types: (a) oneone onto (injective & surjective) (b) oneone into (injective but not surjective) CLASSIFICATION OF FUNCTIONS : One  One Function (Injective mapping) : A function f : A  B is said to be a oneone function or injective mapping if different elements of A have different f images in B . Thus for x1, x2  A & f(x1) ,  B , f(x1) = f(x2)  x1 = x2 or x1  x2 (c)  f(x1)  f(x2) . f(x2) (d) Many–one function : A function f : A  B is said to be a many one function if two or more elements of A have the same f image in B . Thus f : A  B is many one if for ; x1, x2 f(x2) but x1  x2 . manyone onto (surjective but not injective) manyone into (neither surjective nor injective) Identity function : The function f : A  A defined by f(x) = x  x  A is called the identity of A and is denoted by I A. It is easy to observe that identity function is a bijection .  A , f(x1) = Constant function : A function f : A  B is said to be a constant function if every element of A has the same f Onto function (Surjective mapping) : If the function f :AB is such that each image in B . Thus f : A  A , c  B is a constant function. Note that element in B (codomain) is the f image of atleast one element in A, then we say that f is a function of A 'onto' B . Thus f : A  B is surjective if  b that f (a) = b .  B,  some a  A such  B ; f(x) = c ,  x the range of a constant function is a singleton and a constant function may be one-one or many-one, onto or into . 5. ODD & EVEN FUNCTIONS : If f (x) = f (x) for all x in the domain of ‘f’ then f is said to be an even function. e.g. f (x) = cos x ; g (x) = x² + 3 . If f (x) = f (x) for all x in the domain of ‘f’ then f is said to be an odd function. e.g. f (x) = sin x ; g (x) = x3 + x.