Why Sets, Relations & Functions Matter in Your ISC Maths
Ever wondered how a phone book knows which number belongs to which name? That matching idea is a function, and it all starts with sets.
💡 In Simple Words: A set is just a collection of distinct objects, like a basket of different fruits. A relation links items from one set to another, like pairing each fruit with its color. A function is a special relation where each fruit gets exactly one color.
What Is a Set? (And How to Write One)
A set is a group of well‑defined, separate items. We write it inside curly braces { }. For example, {2, 4, 6, 8} is the set of even numbers up to 8.
Important terms:
- Element – an individual object in a set, like a single apple in a fruit basket.
- Subset – a set that lives completely inside another set. If A = {1,2,3} and B = {1,2,3,4,5}, then A ⊆ B.
- Universal set – the “big” set that contains everything we’re talking about in a problem, often denoted by U.
- Empty set – a set with no elements, written ∅.
Common Ways to Describe Sets
1. Roster (listing) method: write every element, like {a, e, i, o, u}.
2. Set‑builder notation: describe a rule, e.g., {x | x is a positive integer less than 5} which means the same as {1,2,3,4}.
Relations: Connecting Two Sets
A relation is just a set of ordered pairs. Think of it as a “match‑making” list.
Example: Let A = {1,2,3} and B = {a,b}. The relation R = {(1,a), (2,b), (3,a)} says 1 is paired with a, 2 with b, and 3 with a.
Key ideas:
- Domain – all the first elements of the pairs (the “left side”). In the example, domain = {1,2,3}.
- Range – all the second elements (the “right side”). Here, range = {a,b}.
- Inverse relation – flip each pair, so R⁻¹ = {(a,1), (b,2), (a,3)}.
When Is a Relation a Function?
A function is a special relation where each element of the domain gets exactly one partner in the range. No element can have two different matches.
Look at R above: each number 1,2,3 appears once, so R is a function. If we added (1,b) as well, it would break the rule because 1 would have two outputs.
Functions: The Heart of Modern Maths
Formally, a function f from set X to set Y is written f: X → Y. It tells you how to turn any x in X into a single y in Y.
Examples you’ll see in exams:
- Linear function: f(x) = 2x + 3. Every x gives one y, and the graph is a straight line.
- Quadratic function: f(x) = x² – 4. The graph is a parabola.
- Constant function: f(x) = 5. No matter what x you plug in, you always get 5.
Important Vocabulary
- Image (or output) – the y you get after applying the function, like the “color” of a fruit.
- Pre‑image (or input) – the x you start with.
- One‑to‑one (injective) – different inputs give different outputs. Imagine each fruit gets a unique color.
- Onto (surjective) – every element of the codomain (the set Y you aim for) actually appears as an output.
- Bijective – both one‑to‑one and onto; a perfect pairing, like a dance where everyone has a partner.
Quick Comparison: Sets vs Relations vs Functions
| Concept | What It Is | Typical Notation | Key Property |
|---|---|---|---|
| Set | Collection of distinct objects | {a, b, c} | Order doesn’t matter |
| Relation | Set of ordered pairs linking two sets | R ⊆ A×B | May map one‑to‑many |
| Function | Special relation with single output per input | f: X → Y | Each x has exactly one y |
How to Solve Typical ISC Questions
Step‑by‑step guide:
- Identify the given sets and write them clearly.
- Read the relation description and list the ordered pairs.
- Check the “single output” rule to decide if it’s a function.
- If asked about injective or surjective, test the definitions with examples.
- Use set‑builder or roster notation as the question demands.
📝 Likely Exam Questions
- Define a function and give an example that is not one‑to‑one.
Model answer: A function assigns exactly one output to each input. Example: f(x)=x² defined on ℝ (real numbers) is not one‑to‑one because f(2)=f(−2)=4. - Let A={1,2,3} and B={a,b,c}. If R={(1,a),(2,b),(3,c)}, find the inverse relation R⁻¹.
Model answer: R⁻¹={(a,1),(b,2),(c,3)}. - State whether the relation S={(x,y) | y = x+1, x∈ℤ} is a function from ℤ to ℤ. Justify.
Model answer: Yes, because for each integer x there is exactly one integer y = x+1. - Determine if the function f:{1,2,3,4}→{a,b,c,d} defined by f(1)=a, f(2)=b, f(3)=c, f(4)=c is onto.
Model answer: No, because element d in the codomain never appears as an output. - Write the set {x | x is a multiple of 3 and 0Model answer: {3,6,9,12}.