Set Notation And Sets Of Numbers
Master set notation and sets of numbers for Queensland Year 11 Mathematical Methods (QCAA). A set is simply a collection of objects, and this topic gives you the language to describe collections of numbers precisely.
You will learn the natural, integer, rational and real number sets, test membership and subsets, take the union and intersection, and switch between set-builder and interval notation — the foundation for stating a domain.
Every question with a fully worked solution.
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Theory
In Year 11 Mathematical Methods (QCAA, Unit 1) the number sets \(\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\) give the language for describing which values a variable may take. This page covers set notation, membership, subsets, union and intersection, and converting between set-builder and interval notation on the number line.
The natural numbers \(\mathbb{N}=\{1,2,3,\dots\}\), the integers \(\mathbb{Z}=\{\dots,-1,0,1,\dots\}\), the rational numbers \(\mathbb{Q}\) (any \(\dfrac{p}{q}\) with integers \(p,q\) and \(q\neq0\)) and the real numbers \(\mathbb{R}\) form the chain \(\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\). A number is irrational (such as \(\sqrt{3}\) or \(\pi\)) when it is real but not rational.
We write \(x\in A\) for “\(x\) is an element of \(A\)” and \(x\notin A\) for “not an element”. The union \(A\cup B\) collects everything in either set; the intersection \(A\cap B\) keeps only what is in both. Over a universal set the complement \(A'\) is everything not in \(A\).
Interval notation uses a square bracket for an included endpoint and a round bracket for an excluded one: \([a,b]\) includes both ends, \((a,b)\) excludes both, and \(\infty\) always takes a round bracket. The equivalent set-builder form is \(\{x:a\le x<b\}\).
Set-builder and interval notation describe the same set of reals:
Union and intersection of sets:
Complement over the reals:
Converting set-builder to interval notation
- Read the inequality inside the braces to find the lower and upper bounds.
- Bracket each end: \(\le\) or \(\ge\) gives a square bracket (closed), \(<\) or \(>\) gives a round bracket (open); use \(\infty\) with a round bracket if unbounded.
- Combine with \(\cup\) if the set is in two pieces, then check the endpoints on a number line.
Simplify first, then place each number:
| \(\sqrt{16}\) | \(=\) | \(4\in\mathbb{N}\) |
| \(-7\) | \(=\) | \(-7\in\mathbb{Z}\) |
| \(\dfrac{2}{5}\) | \(=\) | \(\tfrac{2}{5}\in\mathbb{Q}\) |
| \(\sqrt{3}\) | \(\approx\) | \(1.732\ldots\in\mathbb{R}\) |
\(\sqrt{16}=4\) is a counting number; \(-7\) is a whole number but negative; \(\tfrac{2}{5}\) is a ratio of integers; \(\sqrt{3}\) is irrational, so it is real but not rational.
\(\sqrt{16}\in\mathbb{N},\ -7\in\mathbb{Z},\ \tfrac{2}{5}\in\mathbb{Q},\ \sqrt{3}\in\mathbb{R}\).
Read the bounds — lower is \(-2\), upper is \(5\):
| \(-2\le x\) | \(=\) | \(x\ge-2\ \text{(included)}\) |
| \(x<5\) | \(=\) | \(\text{upper }5\ \text{(excluded)}\) |
Bracket each end — \(\le\) is square, \(<\) is round:
| \(\{x:-2\le x<5\}\) | \(=\) | \([-2,\,5)\) |
\(\{x:-2\le x<5\}=[-2,\,5)\).
Intersection — keep only where the bars overlap:
| \(A\cap B\) | \(=\) | \([-1,4)\cap(2,6]\) |
| \(=\) | \((2,\,4)\) |
The overlap starts just after \(2\) (open in \(B\)) and stops just before \(4\) (open in \(A\)).
Union — take everything covered by either bar:
| \(A\cup B\) | \(=\) | \([-1,4)\cup(2,6]\) |
| \(=\) | \([-1,\,6]\) |
\(A\cap B=(2,\,4)\) and \(A\cup B=[-1,\,6]\).
Describe \(B\) — every real greater than \(2\):
| \(B\) | \(=\) | \((2,\,\infty)\) |
Remove from \(A\) everything greater than \(2\):
| \(A\setminus B\) | \(=\) | \([0,5]\ \text{with}\ x>2\ \text{removed}\) |
| \(=\) | \([0,\,2]\) |
The value \(x=2\) stays because \(2\notin B\) (the round bracket excludes it).
\(A\setminus B=[0,\,2]\).
Common pitfalls
Frequently asked questions
What is the difference between the sets N, Z, Q and R?
\(\mathbb{N}\) is the counting numbers, \(\mathbb{Z}\) adds zero and the negatives, \(\mathbb{Q}\) adds all fractions \(\tfrac{p}{q}\), and \(\mathbb{R}\) adds the irrationals such as \(\sqrt{3}\) and \(\pi\). Each is contained in the next: \(\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\).
When do you use a square bracket versus a round bracket?
A square bracket \([\,]\) means the endpoint is included (\(\le\) or \(\ge\)); a round bracket \((\,)\) means it is excluded (\(<\) or \(>\)). Infinity always takes a round bracket.
What is the difference between union and intersection?
The union \(A\cup B\) contains everything that is in \(A\) or \(B\); the intersection \(A\cap B\) contains only what is in \(A\) and \(B\).
How do you write set-builder notation as an interval?
Read the inequality for the lower and upper bounds, then bracket each end (square for \(\le,\ge\); round for \(<,>\)). For example \(\{x:-2\le x<5\}=[-2,5)\).
Is a number like the square root of 2 rational or irrational?
It is irrational, so it is real but not rational: \(\sqrt{2}\in\mathbb{R}\) but \(\sqrt{2}\notin\mathbb{Q}\). It cannot be written as an exact fraction of two integers.
What does the complement of a set mean?
Over the real numbers, the complement \(A'\) is every real number that is not in \(A\). For example if \(A=(-\infty,1]\) then \(A'=(1,\infty)\).