01 · Explore
Cartesian Products of Sets
The foundation of relations lies in the ability to pair elements from different sets in a specific order.
A Cartesian product of two non-empty sets A and B, denoted as A × B, is the set of all possible ordered pairs (a, b) where 'a' belongs to set A and 'b' belongs to set B. The order is crucial; the pair (a, b) is generally not the same as (b, a) unless a = b.
If set A has 'p' elements and set B has 'q' elements, the total number of elements in the Cartesian product A × B is exactly p × q. If either set is empty, the Cartesian product is also an empty set. As per Example 5 on page 4, R × R represents the coordinates of all points in two-dimensional space, while R × R × R represents the coordinates of all points in three-dimensional space.
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Question
If set A = {1, 2} and set B = {3, 4, 5}, how many elements are in A × B?
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02 · Explore
Understanding Relations
A relation is a way of defining a specific link between elements of two sets.
Formally, a relation R from set A to set B is any subset of the Cartesian product A × B. This subset is formed by selecting ordered pairs that satisfy a specific rule or condition. The second element in an ordered pair is called the 'image' of the first element.
Relations can be represented in three main ways: Roster form (listing all pairs), Set-builder form (stating the rule), and Arrow diagrams (visual paths from elements of A to B).
Components of a Relation
- 1
Domain
The set of all first elements of the ordered pairs in the relation R.
- 2
Range
The set of all second elements (images) that are actually paired in R.
- 3
Codomain
The entire set B from which the range is drawn. Note that Range is always a subset of Codomain.
The logical flow of identifying the parts of a mathematical relation.
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Question
If n(A) = 2 and n(B) = 3, what is the total number of possible relations from A to B?
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03 · Explore
The Concept of Functions
A function is a special type of relation where every input has exactly one output.
A relation 'f' from set A to set B is a function if every element of set A has one, and only one, image in set B. This means no two distinct ordered pairs in the function can have the same first element. If (a, b) is in the function, we write f(a) = b.
A 'real-valued function' is one where the range is a subset of real numbers (R). If the domain is also a subset of R, it is called a 'real function'. Functions are often visualized as 'machines' that take an input 'x' and produce a unique output 'f(x)'.
Not all relations are functions. For instance, Example 9 on page 6 shows that while the number of relations from A to B is 2 raised to the power of n(A × B), a specific relation is only a function if every element in the domain is associated with exactly one image. If an element in the domain is connected to more than one image, the relation fails the definition of a function.
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Question
Is the relation R = {(2, 1), (2, 4), (3, 3)} a function?
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04 · Explore
Standard Real Functions and Graphs
Certain functions appear frequently in mathematics and have distinct graphical shapes.
The Identity Function f(x) = x produces a straight line passing through the origin at a 45-degree angle. The Constant Function f(x) = c produces a horizontal line parallel to the x-axis at height 'c'.
Polynomial functions like f(x) = x² (parabola) or f(x) = x³ (cubic curve) are defined for all real numbers. Rational functions are ratios of polynomials, defined everywhere except where the denominator is zero.
Domain of a Rational Function
f(x) = (x² + 3x + 5) / (x² - 5x + 4)
As shown in Example 21 on page 16, a rational function is undefined when the denominator is zero. We factor x² - 5x + 4 into (x - 4)(x - 1). Setting this to zero gives x = 4 and x = 1. Therefore, the domain is all real numbers except 1 and 4, written as R - {1, 4}.
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Question
What is the range of the constant function f(x) = 5?
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05 · Explore
Modulus, Signum, and Greatest Integer Functions
These functions use specific rules to transform real numbers based on their magnitude or sign.
The Modulus Function f(x) = |x| returns the non-negative value of x. It is defined as x if x ≥ 0 and -x if x < 0. Its graph is V-shaped. The Signum Function extracts the sign of a number, returning 1 for positive numbers, -1 for negative numbers, and 0 for zero.
The Greatest Integer Function f(x) = [x] returns the largest integer less than or equal to x. For example, [2.3] = 2 and [-1.5] = -2. Its graph looks like a series of steps, which is why it is often called a step function.
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Question
What is the value of [ -0.5 ]?
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06 · Explore
Algebra of Real Functions
Functions can be added, subtracted, multiplied, and divided just like numbers.
For two functions f and g with a common domain X, we define (f + g)(x) as f(x) + g(x) and (f - g)(x) as f(x) - g(x). Multiplication by a scalar 'k' is defined as (kf)(x) = k · f(x).
The product (fg)(x) is f(x) · g(x). The quotient (f/g)(x) is f(x) / g(x), provided that g(x) is not zero at that point. These operations allow for the construction of complex mathematical models from simpler base functions.
Operating on Functions
f(x) = x², g(x) = 2x + 1. Find (f + g)(x) and (fg)(x).
Following Example 16 on page 14: (f + g)(x) = f(x) + g(x) = x² + 2x + 1. (fg)(x) = f(x) · g(x) = x²(2x + 1) = 2x³ + x². For the quotient f/g, the domain would exclude x = -1/2 because 2(-1/2) + 1 = 0.
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If f(x) = x and g(x) = 2, what is (f/g)(x)?
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Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 13, 14, 19.