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    MB1201 QR MB1201 QR Presentation Transcript

    • In this document you can review important • Definitions • Facts • Formulae • Procedures Class – XII, CBSE MB1201: Relations And Functions Learn Math for Free. Visit www.ganitgurooz.com
    • Define a relation R from a set A to set B. Class – XII,CBSE MB1201: Relations And Functions Important Definitions Topic: Relations Learn Math for Free. Visit www.ganitgurooz.com
    • Define a relation R from a set A to set B. Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Relations A relation R from a set A to set B is a subset of A × B . If R is a relation from a set A to set B and, if , , then w e say that is related to under the relation R and w e express this fact by w ri a b R a b ting as R .a b Class – XII,CBSE MB1201: Relations And Functions
    • Define a relation R in set A. Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: RelationsClass – XII,CBSE MB1201: Relations And Functions
    • Define a relation R in set A. Important Definitions A relation R in a set A is a subset of .A A Learn Math for Free. Visit www.ganitgurooz.com Topic: RelationsClass – XII,CBSE MB1201: Relations And Functions
    • What is empty relation? Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of RelationsClass – XII,CBSE MB1201: Relations And Functions
    • What is empty relation? Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of Relations A relation R in a set A is called , if no element of A is related to any element of A, i.e., . empty relation R A A Class – XII,CBSE MB1201: Relations And Functions
    • What is universal relation? Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of RelationsClass – XII,CBSE MB1201: Relations And Functions
    • What is universal relation? Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of Relations A relation R in a set A is called universal relation, if each element of A is related to every element of A, i.e., R = A × A. Both the empty relation and the universal relation are called as trivial relations. Class – XII,CBSE MB1201: Relations And Functions
    • Define different types of relations in a set. Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of RelationsClass – XII,CBSE MB1201: Relations And Functions
    • Define different types of relations in a set. Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of Relations 1 2 2 1 1 2 1 2 2 3 1 3 1 2 A relation R in a set A is called: Reflexive, if , , for every , Symmetric, if , implies that , , for all , . Transitive, if , and , implies that , , for all , , a a R a A a a R a a R a a R a a R a a R a a R a a a3 .A Class – XII,CBSE MB1201: Relations And Functions
    • What is equivalence relation? Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of RelationsClass – XII,CBSE MB1201: Relations And Functions
    • What is equivalence relation? Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of Relations A relation R in a set A is said to be an if R is reflexive, symmetric and transitive. equivalence relation Class – XII,CBSE MB1201: Relations And Functions
    • Define one-one and many-one function. Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions
    • Define one-one and many-one function. Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of Functions 1 2 1 2 1 2 A function : is defined to be one-one (or injective), if the images of distinct elements of X under are distinct, i.e., for every , , implies . Otherwise, is called many-one. f X Y f x x X f x f x x x f Class – XII,CBSE MB1201: Relations And Functions
    • Define one-one and many-one function. Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions 1 4 2 3 The function and in the above Figure (i) and (iv) are one-one and the function and in Figure (ii) and (iii) are many-one. f f f f
    • Define onto function. Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions
    • Define onto function. Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions A function : is said to be ( ), if every element of Y is the image of some element of X under , i.e., for every , there exists an element in X such that . f X Y onto or surjective f y Y x f x y
    • Define onto function. Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions 3 4 1 2 1 1 The function and in Figure (iii), (iv) are onto and the function in Figure (i) is not onto as elements and in are not the image of any element in under . f f f e f X X f
    • Define bijective function. Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions
    • Define bijective function. Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions A function : is said to be one-one and onto (or bijective), if is both one-one and onto. f X Y f 4 T he function in Figure (iv) is one-one and onto. f
    • Define composition of functions. Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Composition of FunctionsClass – XII,CBSE MB1201: Relations And Functions
    • Define composition of functions. Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Composition of FunctionsClass – XII,CBSE MB1201: Relations And Functions Let : and : be two functions. Then the composition of and , denoted by , is defined as the function : given by , . f A B g B C f g g f g f A C g f x g f x x A   
    • Define composition of functions. Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Composition of FunctionsClass – XII,CBSE MB1201: Relations And Functions
    • What is invertible function? Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Invertible FunctionClass – XII,CBSE MB1201: Relations And Functions
    • What is invertible function? Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Invertible FunctionClass – XII,CBSE MB1201: Relations And Functions 1 A function : is said to be inver- tible, if there exists a function : such that and . The function is called the and is denoted by , here d X Y X f X Y g Y X g f I f g I g inverse of f f I Invertible function :   enotes the identity function on the set X ,and denotes the identity function on the set Y. Y I
    • Define different binary operations on a set. Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Binary OperationsClass – XII,CBSE MB1201: Relations And Functions
    • Define different binary operations on a set. Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Binary OperationsClass – XII,CBSE MB1201: Relations And Functions A binary operation on a set A is a function : . W e denote ( , ), the value of the function on the element (a, b) by . A binary operation on the set A is called commutative, if , for eve A A A a b a b a b b a ry , .a b A
    • Define different binary operations on a set. Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Binary OperationsClass – XII,CBSE MB1201: Relations And Functions A binary operation on the set a is said to be associative if, , , , , . Given a binary operation : an element , if it exists, is called identity for the operation , if , a b c a b c a b c A A A A e A a e a e a a .A
    • Define different binary operations on a set. Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Binary OperationsClass – XII,CBSE MB1201: Relations And Functions Given a binary operation : with the identity element in A, an element is said to be with respect to the operation , if there exists an element in A such that and A A A e a A invertible b a b e b a b 1 is called the inverse and is denoted by . of a a
    • Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of RelationsClass – XII,CBSE MB1201: Relations And Functions .R X X Fact # 1 Empty relation is the relation R in a set X given by
    • Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of RelationsClass – XII,CBSE MB1201: Relations And Functions .R X X Fact # 2 Universal relation is the relation R in a set X given by
    • Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of RelationsClass – XII,CBSE MB1201: Relations And Functions , .a a R a X Fact # 3 Reflexive relation R in X is a relation with
    • Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of RelationsClass – XII,CBSE MB1201: Relations And Functions ,a b R , .b a R Fact # 4 Symmetric relation R in X is a relation satisfying implies
    • Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of RelationsClass – XII,CBSE MB1201: Relations And Functions ,a b R ,b a R , .a c R Fact # 5 Transitive relation R in X is a relation satisfying and implies that
    • Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Equivalence RelationsClass – XII,CBSE MB1201: Relations And Functions Fact # 6 Equivalence relation R in X is a relation which is reflexive, symmetric and transitive.
    • Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Equivalence RelationsClass – XII,CBSE MB1201: Relations And Functions a a X Fact # 7 Equivalence class containing for an equivalence relation R in a set X is the subset of X containing all elements b related to a.
    • Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions :f X Y 1 2 1 2 1 2 , .f x f x x x x x X Fact # 8 A function is one-one (or injective) if
    • Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions :f X Y ,y Y x X .f x y Fact # 9 A function is onto (or surjective) if given any such that
    • Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions :f X Y Fact # 10 A function is one-one and onto (or bijective), if f is both one-one and onto.
    • Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Composition of FunctionsClass – XII,CBSE MB1201: Relations And Functions :f A B :g B C :g f A C .gof x g f x x A Fact # 11 The composition of functions and is the function given by
    • Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Invertible FunctionsClass – XII,CBSE MB1201: Relations And Functions :f X Y :g Y X X gof I .Y fog I Fact # 12 A function is invertible if such that and
    • A function is invertible if and only if f is one-one and onto. That is “f is invertible, then f must be one-one and onto and conversely, if f is one-one and onto, then f must be invertible”. This fact significantly helps for proving a function f to be invertible by showing that f is one-one and onto, specially when the actual inverse of f is not to be determined. Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Invertible FunctionsClass – XII,CBSE MB1201: Relations And Functions :f X Y Fact # 13
    • Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions :f X X Fact # 14 Given a finite set X, a function is one-one (respectively onto) if and only if f is onto (respectively one-one). This is the characteristic property of a finite set. This is not true for infinite set.
    • Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Binary OperationsClass – XII,CBSE MB1201: Relations And Functions A A Fact # 15 A binary operation on a set A is a function from to A.
    • Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Binary OperationsClass – XII,CBSE MB1201: Relations And Functions e X : ,X X .a e a e a X Fact # 16 An element is the identity element for binary operation If
    • An element is invertible for binary operation if there exists such that where, e is the identity for the binary operation The element b is called inverse of and is denoted by . Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Binary OperationsClass – XII,CBSE MB1201: Relations And Functions a X : ,X X X b X a b e b a . 1 a Fact # 17 a
    • Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Binary OperationsClass – XII,CBSE MB1201: Relations And Functions ,a b b a a b Fact # 18 An operation on X is commutative if in X.
    • Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Binary OperationsClass – XII,CBSE MB1201: Relations And Functions An operation on X is associative if , , in X.a b c a b c a b c Fact # 19
    • Fact # 20 Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Equivalence RelationsClass – XII,CBSE MB1201: Relations And Functions Given an arbitrary equivalence relation R in an arbitrary set X, R divides X into mutually disjoint subsets called partitions or subdivisions of X satisfying the following: All elements of are re i i A A lated to each other, for all . No element of is related to any element of , .i j i A A i j
    • Fact # 20 Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Equivalence ClassesClass – XII,CBSE MB1201: Relations And Functions and , . The subsets are called . j i j i A X A A i j A equivalence classes
    • Fact # 21 Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Equivalence RelationsClass – XII,CBSE MB1201: Relations And Functions If R and S are two equivalence relations on a set A, then is also an equivalence relation on A. R S
    • Fact # 22 Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Equivalence RelationsClass – XII,CBSE MB1201: Relations And Functions The union of two equivalence relations on a set is not necessarily an equivalence relation on the set.
    • Fact # 23 Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Equivalence RelationsClass – XII,CBSE MB1201: Relations And Functions 1 1 If R is an equivalence relation on a set A, then defined as , : , is also an equivalence relation on A . R R b a a b R
    • Fact # 24 Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions For an arbitrary finite set X, a one-one function : is necessarily onto and an onto map : is necessarily one-one. For an infinite set, this may not be true. In fact, this is a characteristic d f X X f X X ifference between a finite and an infinite set.
    • Fact # 25 Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Identity Element for the OperationsClass – XII,CBSE MB1201: Relations And Functions Zero is the identity for the addition operation on R but it is not identity for the addition operation on N, as 0 . In fact the addition operation on N does not have any identity. N
    • Fact # 26 Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Identity Element for the OperationsClass – XII,CBSE MB1201: Relations And Functions For the addition operation : , given any , there exists in R such that a + = 0 (identity for '+') . a a a a a R R R R
    • Fact # 27 Important Facts Learn Math for Free. Visit www.ganitgurooz.com Topic: Identity Element for the OperationsClass – XII,CBSE MB1201: Relations And Functions For the multiplication operation on R, given any 0 in R, we can choose 1 1 1 in R such that 1 (identity for '×') = . a a a a a a
    • Important Formulae If : , : and are functions, then . f X Y g Y Z Z S h g f h g f    Learn Math for Free. Visit www.ganitgurooz.com Topic: Composition of FunctionsClass – XII,CBSE MB1201: Relations And Functions
    • Important Formulae 1 1 1 If : and : be two invertible functions. Then is also invertible with . f X Y g Y Z g f g f f g    Learn Math for Free. Visit www.ganitgurooz.com Topic: Composition of FunctionsClass – XII,CBSE MB1201: Relations And Functions
    • Important Procedures Learn Math for Free. Visit www.ganitgurooz.com Topic: RelationsClass – XII,CBSE MB1201: Relations And Functions Let A be the set of all straight lines in the plane. Define a relation 'R' on A.
    • Important Procedures Learn Math for Free. Visit www.ganitgurooz.com Topic: RelationsClass – XII,CBSE If A is the set of all straight lines in the plane. A relation 'R' on A can be defined as follows: or , if and only if a is parallel to be. W e can easily verify that this relation is an equivale aRb a b R nce relation on A. Thus, we can define its equivalence class denoted by as follows: : , . a A a a b A a b R MB1201: Relations And Functions Let A be the set of all straight lines in the plane. Define a relation 'R' on A.
    • Important Procedures Learn Math for Free. Visit www.ganitgurooz.com Topic: RelationsClass – XII,CBSE W e note the following , , or . For all , , the union of all equivalence classes of elements of A is equal to A. a a A a b A a b a b a A a   MB1201: Relations And Functions Let A be the set of all straight lines in the plane. Define a relation 'R' on A.
    • How to use operation table for the binary operation? Important Procedures Learn Math for Free. Visit www.ganitgurooz.com Topic: Operation Table for the OperationClass – XII,CBSE MB1201: Relations And Functions
    • How to use operation table for the binary operation? Important Procedures Learn Math for Free. Visit www.ganitgurooz.com Topic: Operation Table for the OperationClass – XII,CBSE W hen number of elements in a set A is sm all, we can express a binary operation on the set A with the help of a table called the operation table for the operation . For example consider A = {1, 2, 3}. Then, the operation on A defined as : given by ( , ) max { , } can be expressed by the following operation table Here, (1, 3) = 3, (2, 3) = 3, (1, 2) = 2. A A A a b a b MB1201: Relations And Functions 3* 1 2 31 1 2 32 2 2 3 3 3 3