Class 11th Math Chapter 2: Relations and Functions
Introduction
Relations and functions form a fundamental part of mathematics and are essential for higher studies. They help us understand the connections between different objects or quantities and how they interact with each other. This chapter explores the Cartesian product of sets, relations, and functions in detail.
Ordered Pair
➥ Let 'A' and 'B' be two non-empty sets. If a ∈ A and b ∈ B, then (a, b) is called an ordered pair.
➥ It is denoted by (a, b), where 'a' is the first element and 'b' is the second element.
Equality of Two Ordered Pairs
➥ Two ordered pairs (a1, b1) and (a2, b2) are said to be equal if and only if a1 = a2 and b1 = b2.
Example:
(2, 5) = (2, 5) but (2, 5) ≠ (5, 2)
Cartesian Product of Two Sets
➥ The Cartesian product of two sets A and B, denoted by A × B, is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B. It is defined as:
A × B = { (a, b) | a ∈ A and b ∈ B }
Example:
If A = {1, 2} and B = {x, y}, then the Cartesian product A × B is:
A × B = { (1, x), (1, y), (2, x), (2, y) }
NCERT Example and Solution:
Let A = {1, 2} and B = {a, b}. Find A × B.
Solution:
A × B = { (1, a), (1, b), (2, a), (2, b) }
Cartesian Product of Three Sets
➥ The Cartesian product of three sets A, B, and C, denoted by A × B × C, is the set of all ordered triples (a, b, c) where a ∈ A, b ∈ B, and c ∈ C. It is defined as:
A × B × C = { (a, b, c) | a ∈ A, b ∈ B, c ∈ C }
Example:
If A = {1}, B = {x, y}, and C = {α, β}, then the Cartesian product A × B × C is:
A × B × C = { (1, x, α), (1, x, β), (1, y, α), (1, y, β) }
Relation
➥ Let A and B be two non- empty sets, then any subset of cartesian product A x B is called a relation.
➥ It is a collection of ordered pairs (a, b) where a ∈ A and b ∈ B.
Example:
Let A = {1, 3, 5} and B = {3, 5, 7}
Let Rbe a relation defined on set A into set B such that (a, b) ∈ R; a+b <7 where a ∈ A and b ∈ B.
Then R = { (1, 3), (1, 5), (3, 3) }
NCERT Example and Solution:
Given A = {1, 2} and B = {a, b, c}, find a relation R from A to B such that R = {(1, a), (2, b)}.
Solution:
R = {(1, a), (2, b)}
Domain and Range of the Relation
➥ The domain of a relation R is the set of all first elements of the ordered pairs in R. The range of a relation R is the set of all second elements of the ordered pairs in R.
Function
➥ A function f from a set A to a set B is a relation where every element of A is associated with exactly one element of B. It is denoted as f: A → B.
Example:
Let A = {1, 2, 3} and B = {4, 5, 6}. A function f from A to B can be:
f = { (1, 4), (2, 5), (3, 6) }
Here, each element in A is paired with a unique element in B.
NCERT Example and Solution:
Let A = {1, 2, 3} and B = {a, b, c}. Define a function f: A → B such that f = { (1, a), (2, b), (3, c) }.
Solution:
The function f maps each element in A to a unique element in B as follows:
f(1) = a, f(2) = b, f(3) = c
Domain, Co-domain, and Range of Functions
➥ The domain of a function f is the set of all input values for which the function is defined. The co-domain is the set of all possible output values. The range is the set of actual output values.
Types of Functions
1. Real-Valued Function➥ A function f: A → B is called a real valued function if B is a sunset of real numbers.
Consider the function f(x) = x², where x ∈ ℝ (all real numbers). Here, the range of f(x) is { y ∈ ℝ | y ≥ 0 }, which is a subset of real numbers.
➥ An identity function f: R → R given by f(x) = R for each x ∈ R is called an identity function.
Let A = {1, 2, 3}. The identity function f(x) = x maps each element in A to itself: f(1) = 1, f(2) = 2, f(3) = 3.
➥ A function f: R → R given by f(x) = C for each x ∈ R, where C is constant positive, negative or Zero.
➥ A constant function assigns the same value to every element in its domain.
Consider the function f(x) = 5, where x can be any element in its domain. Here, f(x) always gives the output 5 regardless of the input.
➥ A function f: R → R given by :
f(x) = |x|
➥ It is also known as absolute value function
For x = -3, |x| = 3. For x = 2, |x| = 2. The modulus function always returns a non-negative value.
➥A function f: ℝ → ℝ given by
f(x) = a0 + a1x + a2x2 + ... + anxn,
where n is a non-negative integer and a0, a1, a2, ..., an ∈ ℝ.
Consider the function f(x) = 2x² - 3x + 1. This is a polynomial function where x is raised to non-negative integer powers (2, 1, 0).
➥ A function of the type f(x) g(x) where f(x) and g(x) are polynomial function of x defined in the domain and where g(x) ≠ 0.
Consider the function f(x) = (x² - 4) / (x + 2). This is a rational function where both the numerator and denominator are polynomials.
➥ The signum function, denoted as sgn(x), returns -1 if x < 0, 0 if x = 0, and 1 if x > 0.
sgn(-5) = -1, sgn(0) = 0, sgn(3) = 1. The signum function indicates the sign of a number.
➥ The greatest integer function, denoted as ⌊x⌋ or floor(x), gives the largest integer less than or equal to x.
⌊3.5⌋ = 3, ⌊-2.3⌋ = -3. The greatest integer function rounds down to the nearest integer.
➥ The least integer function, denoted as ⌈x⌉ or ceil(x), gives the smallest integer greater than or equal to x.
⌈3.5⌉ = 4, ⌈-2.3⌉ = -2. The least integer function rounds up to the nearest integer.
