ISC Class 11 Maths: Sets, Relations & Functions Explained

Ever wondered how a simple list of numbers can turn into a rule that predicts every output? Sets, relations, and functions are the language behind that magic.

💡 In Simple Words: A set is a bag of different things. A relation is a list that connects items from one bag to items in another. A function is a special relation where each item in the first bag points to exactly one item in the second bag.

What is a Set?

A set is just a collection of distinct objects, called elements. Think of a set like a basket that holds different fruits – each fruit appears only once, even if you put two apples in, you still count it as one type of fruit in the basket.

We write a set by listing its elements between curly braces: {1, 2, 3} means the set contains the numbers 1, 2 and 3. If a set contains everything we are interested in, we call it the universal set. A set that is completely inside another set is a subset.

Understanding Relations

A relation links elements of one set to elements of another (or the same) set. Imagine you have a list of students and a list of their favourite sports; the relation pairs each student with a sport. Formally, a relation is a set of ordered pairs – a pair written as (a, b) where the first item comes from the first set and the second from the second set.

For example, let A = {Tom, Maya} and B = {Cricket, Chess}. The relation R = {(Tom, Cricket), (Maya, Chess)} tells us who likes which game.

Functions: The Special Kind of Relation

A function is a relation with an extra rule: every element of the first set (called the domain) must be paired with exactly one element of the second set (called the codomain). In everyday language, think of a vending machine – you press one button (input) and you get exactly one snack (output). You can’t press a button and get two different snacks at the same time.

If we write f : X → Y, we read it as “f maps X to Y”. The set of actual outputs that actually appear is the range (sometimes called the image).

Worked Example: From Set to Function

Suppose X = {1,2,3} and we define f(x) = x². List the ordered pairs and state the domain, codomain, and range.

1. Form the pairs: (1,1), (2,4), (3,9). So f = {(1,1), (2,4), (3,9)}.
2. Domain = X = {1,2,3} – the inputs we started with.
3. Codomain is the set of all possible outputs we could have allowed, often taken as the set of real numbers ℝ.
4. Range = {1,4,9} – the outputs that actually show up.

Key Differences at a Glance

Quick Summary

• A set is a simple collection of distinct items.
• A relation pairs items from one set with items of another, written as ordered pairs.
• A function is a relation where every input has one and only one output.
• Domain = all allowed inputs; codomain = all possible outputs; range = outputs that actually occur.

📝 Likely Exam Questions

1. Define a function and give an example with its domain, codomain and range.
   Answer: A function assigns each element of a domain exactly one element of a codomain. Example: f:{1,2,3}→ℝ, f(x)=2x. Domain={1,2,3}, codomain=ℝ, range={2,4,6}.
2. State whether the relation R = {(1,2),(1,3),(2,4)} is a function. Justify.
   Answer: No, because the element 1 from the domain is paired with two different outputs (2 and 3), violating the “one output per input” rule.
3. If A = {a,b,c} and B = {1,2}, list all possible relations from A to B.
   Answer: Any subset of the Cartesian product A×B = {(a,1),(a,2),(b,1),(b,2),(c,1),(c,2)}. There are 2&sup6; = 64 possible relations.
4. Find the range of the function f : ℝ → ℝ defined by f(x)=x²−4.
   Answer: Since x² is always ≥0, x²−4 ≥ −4. The smallest value is −4 (when x=0). Hence range = [−4, ∞).
5. Explain the difference between codomain and range with a short example.
   Answer: Codomain is the set we declare as possible outputs; range is the subset that actually appears. For f:{1,2,3}→ℝ, f(x)=x², codomain = ℝ, range = {1,4,9}.