Set notation
Master set notation for Year 11 Specialist Mathematics in Queensland (QCAA). Set notation is the compact language for describing collections of objects, from the number systems to the sets that appear throughout proof and probability.
You will learn to read and write sets in roster and set-builder form, test membership, count elements with cardinality, and work with subsets, unions, intersections and complements — the foundation for the nature of proof later in the course.
Theory
Set notation is the compact language for describing collections of objects in Year 11 Specialist Mathematics (QCAA, Queensland). It names the number systems \(\mathbb{Z},\mathbb{Q},\mathbb{R}\), records membership, cardinality and subsets, and underpins the language of proof. This page shows how to read and write sets, with worked examples.
A set is a collection of distinct objects called elements. A set can be written in list (roster) form, listing the elements inside braces such as \(A=\{2,4,6,8\}\), or in set-builder form \(A=\{x : x \text{ is even},\ 0 Write \(a\in A\) for "\(a\) is an element of \(A\)" and \(a\notin A\) for "is not an element of". The cardinality \(|A|\) is the number of elements. Two special sets appear often: the empty set \(\varnothing\) (no elements) and the universal set \(\xi\) (everything under discussion). Set \(A\) is a subset of \(B\), written \(A\subseteq B\), when every element of \(A\) is also in \(B\). If in addition \(A\neq B\) it is a proper subset \(A\subset B\). The empty set is a subset of every set. The standard number systems are the integers \(\mathbb{Z}\), the positive integers \(\mathbb{Z}^+\), the rationals \(\mathbb{Q}\) (fractions of integers), the irrationals, and the reals \(\mathbb{R}\). They nest as \(\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\), so every integer is rational and every rational is real.
The cardinality counts the distinct elements of a set:
The union, intersection and complement combine sets within a universal set \(\xi\):
The subset relation is defined by membership:
How to work with a set
- Read the form: a roster set already lists its elements; a set-builder set \(\{x : \dots\}\) gives a domain (such as \(\mathbb{Z}\)) and a condition.
- Generate the elements: run through the domain, keep every value that satisfies the condition, and drop any repeats.
- Answer the question: for cardinality count the distinct elements \(|A|\); for membership test \(a\in A\); for a subset check every element of the smaller set lies in the larger.
- Combine if asked: union keeps elements in either set, intersection keeps those in both, and the complement \(A'\) keeps everything in \(\xi\) outside \(A\).
Find the factor pairs of \(12\), collect every factor, then count the distinct elements:
| \(\text{factor pairs}\) | \(:\) | \(1\times12,\ 2\times6,\ 3\times4\) |
| \(A\) | \(=\) | \(\{1,2,3,4,6,12\}\) |
| \(|A|\) | \(=\) | \(6\) |
\(A=\{1,2,3,4,6,12\}\), so \(|A|=6\).
First form the union (elements in either set), then take everything in \(\xi\) outside it:
| \(\xi\) | \(=\) | \(\{1,2,3,4,5,6,7,8\}\) |
| \(A\cup B\) | \(=\) | \(\{1,2,3,4,5,7\}\) |
| \((A\cup B)'\) | \(=\) | \(\xi \setminus (A\cup B)\) |
| \(=\) | \(\{6,8\}\) |
\((A\cup B)'=\{6,8\}\).
Keep only the elements that appear in both sets:
| \(A\) | \(=\) | \(\{1,2,3,4,5,6\}\) |
| \(B\) | \(=\) | \(\{2,4,6,8,10\}\) |
| \(A\cap B\) | \(=\) | \(\{2,4,6\}\) |
\(A\cap B=\{2,4,6\}\).
Start at \(-2\) (because of \(\le\)) and stop before \(3\) (because of \(<\)); list every integer between, then count:
| \(-2\le x\) | \(<\) | \(3\) |
| \(x\) | \(=\) | \(-2,-1,0,1,2\) |
| \(E\) | \(=\) | \(\{-2,-1,0,1,2\}\) |
| \(|E|\) | \(=\) | \(5\) |
\(E=\{-2,-1,0,1,2\}\), so \(|E|=5\).
Common pitfalls
Frequently asked questions
What is set notation?
Set notation is the symbolic language for describing collections. It uses braces for the elements, \(\in\) for membership, \(|A|\) for how many elements there are, and symbols such as \(\subseteq\), \(\cup\) and \(\cap\) to relate and combine sets.
What is the difference between roster form and set-builder form?
Roster form lists the elements directly, for example \(\{2,4,6\}\). Set-builder form gives a rule, for example \(\{x : x \text{ is even},\ 0
What does cardinality mean?
The cardinality \(|A|\) is the number of distinct elements in \(A\). For \(A=\{2,4,6,8,10\}\), \(|A|=5\). Repeated elements are only counted once.
What are the symbols for the number systems?
\(\mathbb{Z}\) is the integers, \(\mathbb{Z}^+\) the positive integers, \(\mathbb{Q}\) the rationals, \(\mathbb{R}\) the reals. They nest as \(\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\), and \(\sqrt{2}\) is real but not rational.
What is the difference between a subset and a proper subset?
A subset \(A\subseteq B\) allows \(A=B\); a proper subset \(A\subset B\) requires \(A\) to be strictly smaller, so it excludes the case \(A=B\).