Class 11 Mathematics (041) · Chapter 2 NotesRelations and Functions

The Cartesian product of two sets A and B, denoted as A × B, is the set of all ordered pairs (a, b) where the first element a belongs to A and the second element b belongs to B. For example, if A = {1, 2} and B = {3, 4}, then A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}. The Cartesian product is crucial for defining relations and functions between sets. The number of elements in the Cartesian product of two sets is the product of the number of elements in each set. If A has p elements and B has q elements, then A × B has pq elements.

• A × B = {(a, b)a ∈ A, b ∈ B}
• If A and B are non-empty, A × B is non-empty
• If either A or B is infinite, A × B is infinite
• A × A × A = {(a, b, c)a, b, c ∈ A}

A relation R from a set A to a set B is a subset of the Cartesian product A × B. It describes a relationship between elements of A and elements of B. The domain of a relation is the set of all first elements of the ordered pairs, and the range is the set of all second elements. The codomain is the set B. Relations can be represented using set-builder notation or roster notation. The number of possible relations from set A to set B is 2^(pq), where p and q are the number of elements in A and B, respectively.

• A relation R from A to B is a subset of A × B
• Domain of R = {a(a, b) ∈ R}
• Range of R = {b(a, b) ∈ R}
• Codomain of R = B
• Number of relations from A to B = 2^(pq)

A function f from a set A to a set B is a special type of relation where each element of A is associated with exactly one element of B. The domain of a function is the set A, and the range is the set of all images of the elements of A. Functions can be represented using tables, graphs, and equations. A real-valued function has a real number as its output, and a real function has both its domain and range as subsets of real numbers. The identity function, constant function, polynomial function, rational function, modulus function, signum function, and greatest integer function are examples of real functions.

• A function f from A to B is a relation where each a ∈ A has exactly one b ∈ B
• Domain of f = A
• Range of f = {bb = f(a) for some a ∈ A}
• Real-valued functionrange is a subset of R
• Real functiondomain and range are subsets of R

Functions can be classified into different types based on their properties. The identity function f(x) = x assigns to each element x in its domain the element x itself. The constant function f(x) = c assigns the same value c to every element in its domain. Polynomial functions are defined by polynomials, and rational functions are ratios of polynomials. The modulus function f(x) = |x| returns the absolute value of x. The signum function returns 1 for positive x, 0 for x = 0, and -1 for negative x. The greatest integer function [x] returns the greatest integer less than or equal to x.

• Identity functionf(x) = x
• Constant functionf(x) = c
• Polynomial functionf(x) = a0 + a1x + a2x^2 + ... + anx^n
• Rational functionf(x) = p(x)/q(x), where p and q are polynomials
• Modulus functionf(x) = |x|
• Signum functionf(x) = {1 if x > 0, 0 if x = 0, -1 if x < 0}
• Greatest integer functionf(x) = [x]

The composition of two functions f and g, denoted as (f ∘ g)(x), is defined as f(g(x)). A function f is invertible if there exists a function g such that (f ∘ g)(x) = x and (g ∘ f)(x) = x for all x in the domain of f and g, respectively. The inverse function g is denoted as f^(-1). The composition of functions allows for the combination of multiple functions into a single function, and invertible functions have the property that they can be reversed to obtain the original input.

• Composition of functions(f ∘ g)(x) = f(g(x))
• Invertible functionf is invertible if there exists g such that f(g(x)) = x and g(f(x)) = x
• Inverse functiong = f^(-1)

The algebra of real functions involves operations such as addition, subtraction, multiplication, and division of functions. The sum of two functions f and g is defined as (f + g)(x) = f(x) + g(x). The difference of two functions is defined as (f - g)(x) = f(x) - g(x). The product of two functions is defined as (f ⋅ g)(x) = f(x) ⋅ g(x). The quotient of two functions is defined as (f / g)(x) = f(x) / g(x), provided g(x) ≠ 0. These operations allow for the manipulation and combination of functions to form new functions with desired properties.

• Addition(f + g)(x) = f(x) + g(x)
• Subtraction(f - g)(x) = f(x) - g(x)
• Multiplication(f ⋅ g)(x) = f(x) ⋅ g(x)
• Division(f / g)(x) = f(x) / g(x), g(x) ≠ 0