Why Sets, Relations and Functions Matter
Ever wondered how a computer matches a username to a password? That's a function at work, and it all starts with sets and relations. Master these ideas and the rest of maths will feel like a puzzle you can solve.
💡 In Simple Words: A set is just a collection of objects, like a basket of apples. A relation tells you how two sets talk to each other, kind of like a friendship list. A function is a special relation where each item in the first set has exactly one partner in the second set.
What is a Set?
A set is a well‑defined group of distinct objects called elements. "Well‑defined" means you can tell whether any object belongs or not. We write sets with curly braces: {1, 2, 3} means the set containing 1, 2 and 3.
Common Set Notations
- Roster form: List every element, e.g.,
{a, b, c}. - Set‑builder form: Describe a rule, e.g.,
{x | x is an even natural number}reads “the set of all x such that x is an even natural number.” - Universal set (U): The big set that contains everything we’re talking about in a problem.
- Empty set (∅): A set with no elements.
Think of a set like a school bag: you can see each book (element) inside, and you know exactly which books are there.
Understanding Relations
A relation links elements of one set to elements of another (or the same) set. Formally, a relation R from set A to set B is a subset of the Cartesian product A×B. The Cartesian product is all possible ordered pairs (a, b) where a∈A and b∈B.
Example: Let A = {1,2,3} and B = {x,y}. The relation R = {(1,x),(2,y)} says 1 is related to x and 2 is related to y.
Types of Relations
| Property | Meaning | Example |
|---|---|---|
| Reflexive | Every element relates to itself (aRa for all a) | {(1,1),(2,2)} |
| Symmetric | If aRb then bRa | {(1,2),(2,1)} |
| Transitive | If aRb and bRc then aRc | {(1,2),(2,3),(1,3)} |
These properties help you decide whether a relation can be a function or have an inverse.
Functions – The Special Kind of Relation
A function f from set A (domain) to set B (codomain) assigns **exactly one** element of B to each element of A. The set of actual outputs is called the range. In symbols, f: A → B.
Key words:
- Domain: All possible inputs.
- Codomain: The set you promise the outputs will live in.
- Range: The outputs that really appear.
Imagine a vending machine: the domain is the buttons you can press, the codomain is every snack it could possibly hold, and the range is the snacks that actually get dispensed when you press a button.
Worked Example: Is This a Function?
Let A = {−1,0,1} and B = {0,1,2}. Relation R = {(−1,0),(0,1),(1,2)}.
Check each element of A:
- −1 appears once and maps to 0.
- 0 appears once and maps to 1.
- 1 appears once and maps to 2.
Since every input has exactly one output, R is a function. Domain = {−1,0,1}, codomain = {0,1,2}, range = {0,1,2}.
One‑to‑One (Injective) and Onto (Surjective) Functions
- Injective (one‑to‑one): Different inputs give different outputs. No two arrows land on the same output.
- Surjective (onto): Every element of the codomain gets hit by at least one arrow.
When a function is both injective and surjective, we call it bijective. Bijective functions have inverses.
How to Find the Inverse of a Bijective Function
Step 1: Write the equation y = f(x).
Step 2: Swap x and y (because you want x in terms of y).
Step 3: Solve for y – this new expression is f⁻¹(x).
Example: f(x) = 3x + 2.
1) y = 3x + 2.
2) x = 3y + 2.
3) x − 2 = 3y ⇒ y = (x − 2)/3.
So f⁻¹(x) = (x − 2)/3.
Quick Comparison: Sets vs Relations vs Functions
- Set: A collection of distinct objects. No ordering, no pairing.
- Relation: A set of ordered pairs linking elements of two sets. Can be many‑to‑many.
- Function: A relation with the rule “one input → one output.”
📝 Likely Exam Questions
- Define a set and give two different notations for the set of all positive multiples of 3 less than 20.
Answer: A set is a well‑defined collection of distinct elements. Roster form: {3,6,9,12,15,18}. Set‑builder form: {x | x = 3k, k∈ℕ, x - State the three properties of a relation and illustrate each with a small example.
Answer: Reflexive – (a,a) for all a; e.g., {(1,1)}. Symmetric – if (a,b) then (b,a); e.g., {(2,5),(5,2)}. Transitive – if (a,b) and (b,c) then (a,c); e.g., {(1,2),(2,3),(1,3)}. - Given f: ℝ → ℝ, f(x) = x² − 4x + 4, determine whether f is injective, surjective, or bijective.
Answer: f(x) = (x‑2)² ≥ 0, so range = [0,∞). Not every real number is hit → not surjective. f(2)=0 and f(4)=4, different inputs give different outputs, but f(2)=f(0)=4, so not injective. Hence f is neither injective nor surjective, thus not bijective. - Find the inverse of the function g(x) = (5 − x)/2.
Answer: y = (5 − x)/2 → swap → x = (5 − y)/2 → multiply → 2x = 5 − y → y = 5 − 2x. So g⁻¹(x) = 5 − 2x. - Explain in your own words why every function is a relation but not every relation is a function.
Answer: A function is just a set of ordered pairs that obeys the “one input, one output” rule, so it fits the definition of a relation (any set of ordered pairs). A relation may pair one input with many outputs or none at all, breaking the function rule.