Sets & Inequalities
Welcome to Sets & Inequalities — the language mathematicians use to talk about collections of things and to describe ranges of numbers. If you have ever sorted your playlist into folders or picked only the songs you and your friend both like, you already understand the core idea behind sets.
In this module you will learn how to describe sets with proper notation, combine them with union and intersection, visualise them with Venn diagrams, and then move on to inequalities — the rules that tell us which numbers belong inside a given range. Every later topic uses this language, so mastering it early gives you a huge advantage.
By the end you will be comfortable reading expressions like A ∩ B', solving quadratic inequalities, and translating word problems into set or inequality language. Let's dive in!
What the Exam Tests
The official syllabus lists two topics here: sets (definition, operations and representation) and inequalities (their basic properties and how to solve them, quadratic and rational inequalities for example).
Every question on the paper is single-answer multiple choice, set in Chinese or English. Questions in this area can ask you to find an intersection, union or complement, count with a Venn diagram, choose the solution set of an inequality in interval form, or find the values of a parameter for which an inequality holds.
In our mock papers Sets & Inequalities gets 8 of the 48 questions. The CSCA publishes the topics but not how many questions each gets, so this split is our own reading of the syllabus.
A set (集合) is simply a well-defined collection of distinct objects. We usually name sets with capital letters like A, B, C and list their members (called elements, 元素) inside curly braces.
Three properties of elements (人教版): they are definite (确定性 — "the tall students in our class" is not a set, because "tall" has no clear test), distinct (互异性 — no element is listed twice) and unordered (无序性). Distinctness is tested with parameters: for {1, a, a²} to be a set we need a ≠ 1, a ≠ a² and a² ≠ 1, so a ≠ 0, 1, -1. After solving for a parameter, always substitute back and reject any value that makes two elements equal.
Roster (List) Notation
Write every element separated by commas: A = {1, 2, 3, 4, 5}. Order does not matter, and duplicates are ignored — {1, 2, 2, 3} is the same as {1, 2, 3}.
Set-Builder Notation
Describe the rule that elements must satisfy: B = {x | x is an even integer and 0 < x < 10} which equals {2, 4, 6, 8}. The vertical bar "|" is read as "such that".
Key Symbols
∈— "is an element of". Example: 3 ∈ A.∉— "is NOT an element of". Example: 7 ∉ A.∅or{}— the empty set, containing no elements.ℕ= natural numbers {0, 1, 2, …} (自然数集, includes 0),ℕ*orℕ₊= positive integers {1, 2, 3, …} (正整数集),ℤ= integers,ℚ= rationals,ℝ= reals.
Interval Notation (for real-number sets)
[a, b]— closed interval, includes both endpoints.(a, b)— open interval, excludes both endpoints.[a, b)or(a, b]— half-open intervals.(-∞, b]or[a, ∞)— unbounded intervals. Note: ∞ always gets a round bracket because it is not a number you can reach.
Cardinality — the number of elements in a finite set. Written |A| or n(A). For A = {1, 2, 3}, |A| = 3.
Think of set notation as the alphabet of this module — every later topic builds on reading and writing it fluently.
Key takeaway: A set is a collection of distinct elements described by roster notation {1,2,3} or set-builder notation {x | condition}. Master the symbols ∈, ∉, ∅, and interval brackets.
Key Formulas
∈ = element of | ∉ = not element of | |A| = cardinality | ∅ = empty set | Intervals: [a,b] closed, (a,b) open
Worked Example 1
Example 1: Write the set of prime numbers less than 20 in roster notation.
Step 1: List primes: 2, 3, 5, 7, 11, 13, 17, 19.
Step 2: Write: P = {2, 3, 5, 7, 11, 13, 17, 19}.
Step 3: Cardinality: |P| = 8.
Worked Example 2
Example 2: Convert {x ∈ ℤ | -3 ≤ x < 2} to roster notation.
Step 1: We need integers from -3 up to (but not including) 2.
Step 2: List: {-3, -2, -1, 0, 1}.
Worked Example 3
Example 3: Express "all real numbers greater than 5" in interval notation.
Answer: (5, ∞). Round bracket on 5 because "greater than" is strict, round bracket on ∞ always.
Exam tip: Always check whether the problem asks for integers, reals, or natural numbers — the same inequality gives different roster sets for each.
Exam tip: When converting between notations, double-check endpoint inclusion: square bracket = included, round bracket = excluded.
Common mistake: Using square brackets with ∞: Never write [a, ∞] — infinity is not a number you can reach, so always use (a, ∞).
Common mistake: Listing duplicates: {1, 2, 2, 3} is technically {1, 2, 3}. In exam answers keep sets clean with no repeats.
Common mistake: Confusing ∅ with {∅}: The empty set ∅ has 0 elements; {∅} is a set containing the empty set, which has 1 element.
Try This Yourself
Write the set {x ∈ ℕ | x² < 30} in roster notation.
Module Summary
In this module you mastered the language of sets — from basic notation and operations (union, intersection, complement) through Venn diagrams and counting, to subsets and power sets. You then applied these ideas to word problems and moved into inequalities: linear, quadratic, rational, absolute value, and systems. These skills form the foundation for almost every other topic in CSCA Math. Keep practising translating between words, symbols, diagrams, and interval notation — that fluency is what the exam rewards.
Open and read all sections to complete this module