CBSE Class 12 Mathematics Relations And Functions Notes Set 03

RELATIONS AND FUNCTIONS

BASIC CONCEPTS

1. Relation: If \( A \) and \( B \) are two non-empty sets, then any subset \( R \) of \( A \times B \) is called relation from set \( A \) to set \( B \).
   i.e., \( R : A \rightarrow B \Leftrightarrow R \subseteq A \times B \)
   For example: Let \( A = \{1, 2\}, B = \{3, 4\} \)
   Then \( A \times B = \{(1, 3), (1, 4), (2, 3), (2, 4)\} \)
   A subset \( R_1 = \{(1, 3), (2, 4)\} \subseteq A \times B \) is called relation from \( A \) to \( B \).
   Similarly, other subsets of \( A \times B \) are also relation from \( A \) to \( B \).
   If \( (x, y) \in R \), then we write \( x R y \) (read as \( x \) is \( R \) related to \( y \)) and if \( (x, y) \notin R \), then we write \( x \not{R} y \) (read as \( x \) is not \( R \) related to \( y \)).
   
    

2. Domain and Range of a Relation: If \( R \) is any relation from set \( A \) to set \( B \) then,
   (a) Domain of \( R \) is the set of all first coordinates of elements of \( R \) and it is denoted by Dom \( (R) \).
   (b) Range of \( R \) is the set of all second coordinates of \( R \) and it is denoted by Range \( (R) \).
   A relation \( R \) on set \( A \) means, the relation from \( A \) to \( A \) i.e., \( R \subseteq A \times A \).
    
3. Some Standard Types of Relations:
   Let \( A \) be a non-empty set. Then, a relation \( R \) on set \( A \) is said to be
   (a) Reflexive: If \( (x, x) \in R \) for each element \( x \in A \), i.e., if \( x R x \) for each element \( x \in A \).
   (b) Symmetric: If \( (x, y) \in R \)
   \( \implies \) \( (y, x) \in R \) for all \( x, y \in A \), i.e., if \( x R y \)
   \( \implies \) \( y R x \) for all \( x, y \in A \).
   (c) Transitive: If \( (x, y) \in R \) and \( (y, z) \in R \)
   \( \implies \) \( (x, z) \in R \) for all \( x, y, z \in A \), i.e., if \( x R y \) and \( y R z \)
   \( \implies \) \( x R z \).
    
4. Equivalence Relation: Any relation \( R \) on a set \( A \) is said to be an equivalence relation if \( R \) is reflexive, symmetric and transitive.
    
5. Antisymmetric Relation: A relation \( R \) in a set \( A \) is antisymmetric
   if \( (a, b) \in R, (b, a) \in R \)
   \( \implies \) \( a = b \), \( \forall \ a, b \in R \), or \( a R b \) and \( b R a \)
   \( \implies \) \( a = b \), \( \forall \ a, b \in R \).
   For example, the relation "greater than or equal to, \( \geq \)" is antisymmetric relation as
   \( a \geq b, b \geq a \)
   \( \implies \) \( a = b \ \forall \ a, b \)
   [Note: "Antisymmetric" is completely different from not symmetric.]
    
6. Equivalence Class: Let \( R \) be an equivalence relation on a non-empty set \( A \). For all \( a \in A \), the equivalence class of '\( a \)' is defined as the set of all such elements of \( A \) which are related to '\( a \)' under \( R \). It is denoted by \( [a] \).
   i.e., \( [a] = \) equivalence class of '\( a \)' \( = \{x \in A : (x, a) \in R\} \)
   For example, Let \( A = \{1, 2, 3\} \) and \( R \) be the equivalence relation on \( A \) given by
   \( R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)\} \)
   The equivalence classes are
   \( [1] = \) equivalence class of \( 1 = \{x \in A : (x, 1) \in R\} = \{1, 2\} \)
   Similarly, \( [2] = \{2, 1\} \) and \( [3] = \{3\} \)
    
7. Function: Let \( X \) and \( Y \) be two non-empty sets. Then, a rule \( f \) which associates to each element \( x \in X \), a unique element, denoted by \( f(x) \) of \( Y \), is called a function from \( X \) to \( Y \) and written as \( f : X \rightarrow Y \) where, \( f(x) \) is called image of \( x \) and \( x \) is called the pre-image of \( f(x) \) and the set \( Y \) is called the co-domain of \( f \) and \( f(X) = \{f(x): x \in X\} \) is called the range of \( f \).
    
8. Types of Function:
   (i) One-one function (injective function): A function \( f : X \rightarrow Y \) is defined to be one-one if the image of distinct element of \( X \) under rule \( f \) are distinct, i.e., for every \( x_1, x_2 \in X \), \( f(x_1) = f(x_2) \) implies that \( x_1 = x_2 \).
   (ii) Onto function (Surjective function): A function \( f : X \rightarrow Y \) is said to be onto function if each element of \( Y \) is the image of some element of \( x \) i.e., for every \( y \in Y \), there exists some \( x \in X \), such that \( y = f(x) \). Thus \( f \) is onto if range of \( f = \) co-domain of \( f \).
   (iii) One-one onto function (Bijective function): A function \( f : X \rightarrow Y \) is said to be one-one onto, if \( f \) is both one-one and onto.
   (iv) Many-one function: A function \( f : X \rightarrow Y \) is said to be a many-one function if two or more elements of set \( X \) have the same image in \( Y \). i.e.,
   \( f : X \rightarrow Y \) is a many-one function if there exist \( a, b \in X \) such that \( a \neq b \) but \( f(a) = f(b) \).
    
9. Composition of Functions: Let \( f : A \rightarrow B \) and \( g : B \rightarrow C \) be two functions. Then, the composition of \( f \) and \( g \), denoted by \( gof \), is defined as the function.
    
10. Identity Function: Let \( R \) be the set of real numbers. A function \( I : R \rightarrow R \) such that
    \( I (x) = x \ \forall \ x \in R \) is called identity function. Obviously, identity function associates each real number to itself.
     
11. Invertible Function: For \( f : A \rightarrow B \), if there exists a function \( g : B \rightarrow A \) such that \( gof = I_A \) and \( fog = I_B \), where \( I_A \) and \( I_B \) are identity functions, then \( f \) is called an invertible function, and \( g \) is called the inverse of \( f \) and it is written as \( f^{-1} = g \).
     
12. Number of Functions: If \( X \) and \( Y \) are two finite sets having \( m \) and \( n \) elements respectively then the number of functions from \( X \) to \( Y \) is \( n^m \).
     
13. Vertical Line Test: It is used to check whether a relation is a function or not. Under this test, graph of given relation is drawn assuming elements of domain along \( x \)-axis. If a vertical line drawn anywhere in the graph, intersects the graph at only one point then the relation is a function, otherwise it is not a function.
     
14. Horizontal Line Test: It is used to check whether a function is one-one or not. Under this test graph of given function is drawn assuming elements of domain along \( x \)-axis. If a horizontal line (parallel to \( x \)-axis) drawn anywhere in graph, intersects the graph at only one point then the function is one-one, otherwise it is many-one.
    (a) \( f(x) = 2x + 1 \) is one-one function.
    (b) \( f(x) = x^2 \) is many-one function.