Why Sets, Relations & Functions Matter in Your ISC Maths Exam
Imagine trying to sort your school bag: books go in one pocket, pens in another, and the snack bar stays separate. That simple act is a real‑life version of the set ideas you’ll meet in the exam – and the way you link items (like matching a book to its subject) is exactly what relations and functions do on paper.
💡 In Simple Words: A set is just a collection of distinct objects, a relation tells how two sets are paired, and a function is a special relation where each input has one and only one output.
What Is a Set?
A set is a well‑defined group of things called elements. "Well‑defined" means you can tell whether any object belongs or not. We write a set by listing its elements inside curly braces, like {1,2,3}, or by describing a rule, like {x | x is a natural number less than 5}.
Common Set Notations
- ∈ means “is an element of”. Example:
3 ∈ {1,2,3}. - ∉ means “is not an element of”.
- ⊂ means “is a subset of” (every element of the first set is also in the second).
- U denotes the universal set – the big set that contains everything we are talking about.
- ∅ is the empty set, a set with no elements.
Worked Example: Listing a Set
List the set of all even numbers between 1 and 10.
Solution: The even numbers are 2, 4, 6, 8, and 10, so we write {2,4,6,8,10}. Notice we stopped at 10 because the condition says “between 1 and 10”.
Relations: Connecting Elements
A relation is a rule that pairs elements from one set (called the first set) with elements of another set (the second set). Formally, a relation is a subset of the Cartesian product A×B, where A×B = {(a,b) | a∈A and b∈B}. Think of the Cartesian product as a grid: each cell holds an ordered pair (first element, second element).
How to Write a Relation
Suppose A = {1,2,3} and B = {x,y}. The relation R = {(1,x),(2,y),(3,x)} tells us which number from A goes with which letter from B. Notice the order matters – (1,x) is different from (x,1).
Properties of Relations
| Property | Meaning | Example |
|---|---|---|
| Reflexive | Every element relates to itself (a,a) for all a in the set. | {(1,1),(2,2)} on {1,2} is reflexive. |
| Symmetric | If (a,b) is in the relation, then (b,a) is also in it. | {(1,2),(2,1)} is symmetric. |
| Transitive | If (a,b) and (b,c) are in the relation, then (a,c) must be too. | {(1,2),(2,3),(1,3)} is transitive. |
Worked Example: Checking Reflexive Property
Let S = {1,2,3} and R = {(1,1),(2,2),(3,3),(1,2)}. Is R reflexive on S?
We need (1,1), (2,2) and (3,3). All three are present, so R **is** reflexive, even though it has an extra pair (1,2).
Functions: Special Kind of Relation
A function is a relation with a stricter rule: each input (called the domain) is paired with exactly one output (called the range). The set that contains all possible outputs is the codomain. In everyday language, a function is like a vending machine – you press one button (input) and it gives you one snack (output).
Key Terms
- Domain – the set of all allowed inputs.
- Codomain – the set where outputs live, even if some values never appear.
- Range – the actual set of outputs that occur.
One‑to‑One, Onto and Bijective
- One‑to‑One (Injective): No two different inputs share the same output. Imagine each student gets a unique locker number.
- Onto (Surjective): Every element of the codomain is hit by at least one input. Like every seat in a theater is occupied.
- Bijective: Both one‑to‑one and onto; a perfect pairing, like matching socks after laundry.
Worked Example: Is f(x)=2x+3 a One‑to‑One Function?
Take two arbitrary inputs a and b. If f(a)=f(b), then 2a+3 = 2b+3. Subtract 3 and divide by 2, we get a = b. Since equal outputs force equal inputs, the function is one‑to‑one.
Quick Comparison
- Set: Collection of distinct objects. No ordering, no pairing.
- Relation: Subset of a Cartesian product; can pair many‑to‑many.
- Function: Relation with the rule “each input → exactly one output”.
📝 Likely Exam Questions
- Define a set and give two examples – one using roster method and one using set‑builder notation.
Answer: A set is a well‑defined collection of distinct elements. Roster example:{a,b,c}. Set‑builder example:{x | x is an even natural number ≤ 10}. - State and explain the three properties of a relation. Provide a short example for each.
Answer: Reflexive – (a,a) for all a; e.g., {(1,1)} on {1}. Symmetric – (a,b) ⇒ (b,a); e.g., {(2,3),(3,2)}. Transitive – (a,b) and (b,c) ⇒ (a,c); e.g., {(1,2),(2,3),(1,3)}. - Given the function
g : ℝ → ℝ, g(x)=x², state its domain, codomain, range and determine whether it is one‑to‑one or onto.
Answer: Domain = ℝ, Codomain = ℝ, Range = {x≥0}. Not one‑to‑one (g(2)=g(‑2)). Not onto because negative numbers are never produced. - Explain the difference between “range” and “codomain” with a suitable example.
Answer: Codomain is the set we declare as possible outputs; range is the set actually obtained. Forh:{1,2,3}→{a,b,c}, h(1)=a, h(2)=a, h(3)=b, codomain = {a,b,c}, range = {a,b}. - Prove that the relation
R = {(x,y) | y = x+2}on the set of integers is a function and state whether it is one‑to‑one and onto.
Answer: For each integer x there is exactly one y = x+2, so it is a function. It is one‑to‑one because different x give different y. It is not onto ℤ because no integer maps to, say, 1 when x is any integer? Actually every integer y can be written as x+2 (choose x = y‑2), so it is onto ℤ as well. Hence it is bijective.