Comprehensive theory, key formulas, diagrams, and memory aids for Relations and Functions.
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.
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)}.
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.
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.
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}.
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.
The domain of R^(-1) is the range of R, and the range of R^(-1) is the domain of R.
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.
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.
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.
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}.
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.
f(x) = ax + b, where a is not 0. The graph is a straight line with slope a and y-intercept b.
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.
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.
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.
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}.
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.
If f and g are real functions with domains D1 and D2, then:
| 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 |
| 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)} |
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"]
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.