Sets and set operations

Your class, the songs on your phone, the even numbers: we group things all the time. Mathematics gives such a group a name, a set, and a few simple tools for working with it. Almost every later topic, from equations to probability, is written in this language.

Sets and elements

A set is a collection of distinct objects, called its elements. We write $a \in A$ if $a$ is an element of $A$ and $a \notin A$ if it is not.

There are two ways to describe a set.

• Listing: $A=\{2,4,6,8\}$. The order does not matter and repeating an element changes nothing.
• By a property: $A=\{x \in \mathbb{N} \mid x \text{ is even and } x \le 8,\ x>0\}$. Read "the set of all $x$ in $\mathbb{N}$ such that ...".

The number of elements of a finite set $A$ is written $n(A)$. The empty set $\emptyset$ has no element, so $n(\emptyset)=0$.

Subsets and equal sets

For example $\{1,2\}\subset\{1,2,3\}$. The set $\{1,2,3\}$ has $8$ subsets: $\emptyset$, $\{1\}$, $\{2\}$, $\{3\}$, $\{1,2\}$, $\{1,3\}$, $\{2,3\}$, $\{1,2,3\}$.

Number sets

• $\mathbb{N}=\{0,1,2,3,\dots\}$ natural numbers, and $\mathbb{N}^*=\{1,2,3,\dots\}$ positive integers.
• $\mathbb{Z}$: integers $\dots,-2,-1,0,1,2,\dots$
• $\mathbb{Q}$: rational numbers, those of the form $\dfrac{p}{q}$ with $p,q\in\mathbb{Z}$, $q\neq 0$.
• $\mathbb{R}$: real numbers, all points of the number line.

They are nested: $\mathbb{N}^*\subset\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$. The number $\sqrt{2}$ belongs to $\mathbb{R}$ but not to $\mathbb{Q}$.

Intervals

Many subsets of $\mathbb{R}$ are pieces of the number line. Let $a<b$.

Set	Name	Notation
$\{x \mid a<x<b\}$	open interval	$(a, b)$
$\{x \mid a\le x\le b\}$	closed interval (segment)	$[a, b]$
$\{x \mid a\le x<b\}$	half-open interval	$[a, b)$
$\{x \mid a<x\le b\}$	half-open interval	$(a, b]$
$\{x \mid x>a\}$ and $\{x\mid x\ge a\}$	rays	$(a, +\infty)$ and $[a, +\infty)$
$\{x \mid x<b\}$ and $\{x\mid x\le b\}$	rays	$(-\infty, b)$ and $(-\infty, b]$

A square bracket means the end is included (filled dot), a round bracket means it is excluded (hollow dot). Infinity is never included.

Operations on sets

Operations on intervals

Counting elements

For finite sets, adding $n(A)+n(B)$ counts the elements of $A\cap B$ twice, so we subtract them once:

Key terms

Summary

• A set is described by listing its elements or by a property; $\emptyset$ is the empty set; $A\subset B$ means every element of $A$ is in $B$.
• $\mathbb{N}^*\subset\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$.
• Intervals use square brackets for included ends and round brackets for excluded ends; infinity always gets a round bracket.
• Union means "or", intersection means "and", difference $A\setminus B$ means "in $A$ but not in $B$", complement is $E\setminus A$.
• $n(A\cup B)=n(A)+n(B)-n(A\cap B)$.

Practice

Show solutions

1. (a) $\{2,3,4,5,6\}$. (b) $\{-2,-1,0,1,2\}$. (c) $x^2=9$ gives $x=\pm 3$: $\{-3,3\}$.
2. (a) The odd natural numbers less than $12$: $\{x\in\mathbb{N}\mid x\text{ odd},\ x<12\}$. (b) The positive multiples of $3$ that are at most $18$: $\{x\in\mathbb{N}^*\mid x=3k,\ k\in\mathbb{N}^*,\ x\le 18\}$.
3. (a) True. (b) False, $-3$ is negative. (c) True. (d) False, $\sqrt2$ is irrational. (e) True, every integer $k$ equals $\frac{k}{1}$. (f) False, $0$ is not positive.
4. A subset containing $1$ is $\{1\}$ together with a subset of $\{2,3\}$: $\{1\}$, $\{1,2\}$, $\{1,3\}$, $\{1,2,3\}$. There are $4$ (half of the $8$ subsets).
5. (a) $(-1, 4]$. (b) $[2, +\infty)$. (c) $(-\infty, 0)$.
6. $A\cap B=(0, 2]$. $A\cup B=[-3, 6)$. $A\setminus B=[-3, 0]$ ($0$ is not in $B$). $B\setminus A=(2, 6)$ ($2$ is in $A$, so it is removed).
7. $A=\{0,1,2,3,4,5,6\}$, $B=\{0,2,4,6,8,10\}$. $A\cap B=\{0,2,4,6\}$. $A\cup B=\{0,1,2,3,4,5,6,8,10\}$. $A\setminus B=\{1,3,5\}$. Check: $n(A\cup B)=9$ and $7+6-4=9$.
8. $n(A\cup B)=45$, so $45=30+22-n(A\cap B)$, hence $n(A\cap B)=52-45=7$.
9. The numbers not in $A$ are those with $x>1$ and $x\le 4$: the complement is $(1, 4]$.
10. (a) At least one language: $60+45-20=85$. Exactly one: $85-20=65$ (or $40+25$). Neither: $100-85=15$. (b) We need $m\ge 0$ and $m+2\le 5$, i.e. $0\le m\le 3$.