Relations and functions form the conceptual bridge between sets and the rest of mathematics. A relation connects the elements of two sets according to a rule, while a function is a special kind of relation in which every element of the first set is associated with exactly one element of the second set. This chapter formalises these ideas using ordered pairs and Cartesian products.
The chapter begins with ordered pairs and the Cartesian product of sets, then defines relations, their domain and range, and various types of relations such as empty, universal, identity, and inverse relations. It then introduces functions, discusses their graphical representation, and studies important classes of functions including constant, identity, polynomial, rational, modulus, signum, and greatest integer functions.
Understanding functions is absolutely essential for calculus and for every quantitative science. The concept of a function as a machine that assigns a unique output to every input reappears in limits, derivatives, integration, and in applied fields such as physics, economics, and engineering. This chapter lays the precise groundwork for all of that.
2. Cartesian Product of Sets
For two non-empty sets A and B, the Cartesian product A x B is the set of all ordered pairs (a, b) where a in A and b in B.
A x B = {(a, b) : a in A, b in B}
An ordered pair (a, b) has the property that (a, b) = (c, d) if and only if a = c and b = d. The order matters, which is why these are called ordered pairs.
Example: If A = {1, 2} and B = {x, y}, then A x B = {(1, x), (1, y), (2, x), (2, y)}.
Properties of Cartesian Product
If A and B are finite sets with n(A) = m and n(B) = n, then n(A x B) = mn.
If either A or B is the empty set, then A x B = phi.
In general, A x B is not equal to B x A unless A = B or one set is empty.
A x (B union C) = (A x B) union (A x C).
A x (B intersection C) = (A x B) intersection (A x C).
Cartesian Product with Three Sets
The Cartesian product of three sets A, B, C is A x B x C = {(a, b, c) : a in A, b in B, c in C}. If n(A) = p, n(B) = q, n(C) = r, then n(A x B x C) = pqr.
3. Relations
A relation R from a set A to a set B is a subset of A x B. If (a, b) is in R, we say a is related to b and write a R b.
Domain and Range
The domain of R is the set of all first elements of the ordered pairs in R.
The range of R is the set of all second elements of the ordered pairs in R.
If R is a relation from A to B, then the domain of R is a subset of A and the range of R is a subset of B.
Example: If A = {1, 2, 3} and B = {3, 4, 5}, the relation R = {(1, 3), (2, 3), (3, 4)} has domain {1, 2, 3} and range {3, 4}.
Total Number of Relations
If A has m elements and B has n elements, then A x B has mn elements and the number of relations from A to B is 2^(mn), since each subset of A x B is a relation.
4. Types of Relations
Empty relation: A relation R in a set A such that no element of A is related to any element. Written as R = phi.
Universal relation: R = A x A, meaning every element is related to every element.
Identity relation: I = {(a, a) : a in A}, each element related only to itself.
Inverse relation: If R is a relation from A to B, the inverse relation R^(-1) from B to A is R^(-1) = {(b, a) : (a, b) in R}.
The domain of R^(-1) is the range of R, and the range of R^(-1) is the domain of R.
5. Functions
A relation f from a set A to a set B is called a function if every element of A is associated with a unique element of B. Symbolically, f : A to B means f is a function from A to B.
Domain of f: the set A.
Codomain of f: the set B.
Range of f: the set of all images f(a) for a in A, which is a subset of B.
If f(a) = b, then b is called the image of a, and a is called the pre-image of b.
A function must satisfy two conditions: (i) every element of A has an image, and (ii) no element of A has two different images. A relation that fails either condition is not a function.
Real-Valued Functions
A function f : A to B is real-valued if B is a subset of R, and a function f : A to B is a real function if both A and B are subsets of R. The domain of a real function is the set of all real values of x for which f(x) is a real number.
6. Some Standard Functions and Their Graphs
Constant Function
f(x) = c for all x in R. Its graph is a horizontal straight line parallel to the x-axis. The domain is R and the range is {c}.
Identity Function
f(x) = x for all x in R. The graph is the straight line y = x passing through the origin. Domain and range are both R.
Linear Function
f(x) = ax + b, where a is not 0. The graph is a straight line with slope a and y-intercept b.
Polynomial Function
f(x) = a_0 + a_1 x + ... + a_n x^n, where n is a non-negative integer and a_n is not 0. The domain is R.
Rational Function
f(x) = p(x)/q(x), where p(x) and q(x) are polynomials and q(x) is not the zero polynomial. The domain is R minus the points where q(x) = 0.
Modulus Function
f(x) = |x|, defined as |x| = x if x >= 0, and |x| = -x if x < 0. The domain is R and the range is [0, infinity). Its graph is V-shaped, symmetric about the y-axis.
Signum Function
f(x) = 1 if x > 0, f(x) = 0 if x = 0, f(x) = -1 if x < 0. The domain is R and the range is {-1, 0, 1}.
Greatest Integer Function
f(x) = [x], the greatest integer less than or equal to x. For example, [2.7] = 2, [-1.5] = -2. The domain is R and the range is Z. The graph looks like a staircase.
7. Algebra of Real Functions
If f and g are real functions with domains D1 and D2, then:
Scalar multiple: (c . f)(x) = c . f(x) for any real number c.
Quick Revision Tables
Table 1: Types of Functions and Their Properties
Function
Definition f(x)
Domain
Range
Graph Shape
Constant
c
R
{c}
Horizontal line
Identity
x
R
R
Line y = x
Modulus
x for x >= 0, -x for x < 0
R
[0, infinity)
V-shape
Signum
1, 0, -1
R
{-1, 0, 1}
Three points
Greatest integer
[x]
R
Z
Staircase
Linear
ax + b
R
R
Straight line
Table 2: Relation Versus Function
Feature
Relation
Function
Definition
Any subset of A x B
Subset of A x B where each element of A has a unique image
First element
May be repeated
Each element of A appears exactly once as a first element
Image
May have many
Exactly one image per element
Number
Up to 2^(mn) relations
Limited by uniqueness rule
Example
{(1,2), (1,3)}
{(1,2), (2,3)}
Mind Map
graph TD
A["Relations and Functions"] --> B["Cartesian Product"]
A --> C["Relations"]
A --> D["Functions"]
A --> E["Standard Functions"]
A --> F["Algebra of Functions"]
B --> B1["A x B = {(a, b)}"]
B --> B2["n(A x B) = mn"]
C --> C1["Subset of A x B"]
C --> C2["Domain and Range"]
C --> C3["Types: empty, universal, identity, inverse"]
D --> D1["Each element maps to unique image"]
D --> D2["Domain, codomain, range"]
E --> E1["Constant, Identity, Modulus"]
E --> E2["Signum, Greatest integer"]
F --> F1["(f + g)(x) = f(x) + g(x)"]
F --> F2["(f/g)(x) = f(x)/g(x), g(x) not = 0"]
Important Diagrams (SVG)
Diagram 1: Arrow Diagram of a Function and a Relation
Diagram 2: Graphs of Modulus, Identity and Constant Functions
Common Mistakes
Confusing the Cartesian products A x B and B x A. They are equal only when A = B or one of them is the empty set, because the order in an ordered pair matters.
Writing (a, b) = (b, a). An ordered pair respects order; (1, 2) is not equal to (2, 1).
Using n(A x B) = n(A) + n(B). The correct formula is n(A x B) = n(A) x n(B).
Calling a relation a function when some element of the domain has two different images. Uniqueness of image is essential.
Confusing range with codomain. The range is a subset of the codomain; it contains only the actual images.
For the modulus function, writing |x| = x for all x. When x < 0, |x| = -x.
For the greatest integer function, computing [-1.5] as -1. The correct value is -2, since -2 is the greatest integer less than or equal to -1.5.
Dividing functions without excluding the points where g(x) = 0. The domain of f/g excludes those points.
Forgetting that the signum function is not defined at x = 0 by the usual formula; it is specially defined to have value 0 at x = 0.
Exam Tips
When asked to count the number of relations from A to B, compute 2^(n(A) x n(B)), not 2^(n(A) + n(B)).
To prove a relation is a function, check that every element of the domain appears exactly once as a first element and that no domain element maps to two values.
For the modulus function, split into two cases x >= 0 and x < 0, and always check the answer is non-negative.
The greatest integer function is a favourite; remember that for negative numbers you move to the lower integer, for example [2.7] = 2 but [-2.7] = -3.
When computing the domain of a rational function, solve q(x) = 0 and exclude those values from R.
Practise drawing the graphs of modulus, signum, and greatest integer functions, since graph-based questions appear frequently.
For algebra of functions, note that all operations except division keep the domain as the intersection of individual domains; division additionally removes points where the denominator is zero.
Conclusion
Relations and functions give mathematics a precise language for describing how quantities depend on one another. The Cartesian product organises pairs of objects, relations generalise the idea of connection, and functions capture the crucial property of unique association between inputs and outputs. The standard functions studied here, including modulus, signum, and greatest integer functions, appear constantly in limits, continuity, and differentiation. Mastering the definitions of domain, codomain, range, and the algebra of functions will make the study of calculus and of applied mathematics dramatically smoother, and it provides the essential vocabulary used throughout the rest of the Class 11 and Class 12 syllabus.