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. These skills appear in almost every CSCA Math section, so mastering them 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!
The CSCA exam typically includes 3-5 questions on sets and inequalities, worth a total of 12-20 marks. Expect: one Venn diagram problem (2 or 3 circles), one or two inequality-solving questions (quadratic or absolute value), and a word problem that combines sets with counting. The key skill tested is translating between words, notation, diagrams, and interval notation.
A set 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) inside curly braces.
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, …}, 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.
💡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.
🧠Always check whether the problem asks for integers, reals, or natural numbers — the same inequality gives different roster sets for each.
🧠When converting between notations, double-check endpoint inclusion: square bracket = included, round bracket = excluded.
⚠️Using square brackets with ∞: Never write [a, ∞] — infinity is not a number you can reach, so always use (a, ∞).
⚠️Listing duplicates: {1, 2, 2, 3} is technically {1, 2, 3}. In exam answers keep sets clean with no repeats.
⚠️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.
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, 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