Sets, Relations and Functions: Class 11 Basics
Sets, relations and functions form the foundation of higher mathematics and are one of the first big topics in Class 11. Once you understand them, calculus, probability and coordinate geometry all become much easier. This guide explains the notation and types with simple examples aimed at CBSE and JEE students.
What is a set?
A set is a well-defined collection of distinct objects, called elements. We write sets using curly brackets. For example, A = {1, 2, 3, 4, 5} is the set of the first five natural numbers. If 3 belongs to A we write 3 ∈ A; since 7 does not, we write 7 ∉ A.
- Empty set: a set with no elements, written ∅ or { }.
- Finite set: has a countable number of elements, like {a, b, c}.
- Infinite set: has unlimited elements, like the set of all natural numbers N.
- Subset: B ⊆ A means every element of B is also in A.
Operations on sets
The main operations are union, intersection and difference. Let A = {1, 2, 3} and B = {2, 3, 4}.
- Union (A ∪ B): all elements in either set = {1, 2, 3, 4}.
- Intersection (A ∩ B): elements common to both = {2, 3}.
- Difference (A − B): in A but not B = {1}.
Cartesian product and relations
The Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B. If A = {1, 2} and B = {x, y}, then A × B = {(1, x), (1, y), (2, x), (2, y)}.
A relation R from A to B is simply a subset of A × B. For example, on the set {1, 2, 3}, the relation "is less than" gives R = {(1, 2), (1, 3), (2, 3)}. Relations connect elements according to a rule, and functions are a very special type of relation.
What is a function?
A function f from set A to set B assigns to every element of A exactly one element of B. Here A is the domain, B is the codomain, and the set of actual output values is the range. The key rule: no input can have two outputs.
Example: Let f(x) = 2x + 1 with domain {1, 2, 3}. Then f(1) = 3, f(2) = 5, f(3) = 7, so the range is {3, 5, 7}. Because each input gives exactly one output, this is a valid function.
Types of functions
- One-one (injective): different inputs give different outputs. f(x) = 2x + 1 is one-one.
- Onto (surjective): every element of the codomain is an output.
- Bijective: both one-one and onto — this allows an inverse function.
- Many-one: at least two inputs share an output, like f(x) = x², where f(2) = f(−2) = 4.
These ideas are used heavily in Class 11 Maths and continue into Class 12 Maths, where inverse and composite functions appear. Strong notation habits also help with symbols and formulas, so our tips on how to memorise maths formulas are worth a read.
Frequently asked questions
Is every relation a function?
No. Every function is a relation, but not every relation is a function. A relation becomes a function only when each input from the domain is linked to exactly one output.
What is the difference between codomain and range?
The codomain is the set in which outputs are allowed to lie, while the range is the set of outputs actually produced by the function. The range is always a subset of the codomain.
How many elements are in A × B?
If set A has m elements and set B has n elements, then A × B has exactly m × n ordered pairs. For example, a 2-element set and a 3-element set give 6 pairs.